This chapter points out the limitations in simulating quantum systems in classical computers. It describes the various stages of quantum simulation using quantum computers and relevant quantum circuits. Quantum simulators are classified into two types: analog quantum simulators and digital quantum simulators. In an analog quantum simulator the physical properties of a targeted model are reproduced on a physical set-up under externally controlled conditions. Quantum simulation will be considered successful only if the initial state preparation, the implementation of the time evolution and the measurements of observables are realized using only polynomial resources. The chapter discusses the quantum circuits for measurement of the dynamical correlation functions and the spectrum of the Hermitian operators. Computing the time evolution operator is equivalent to the task of implementing a unitary matrix and any unitary transformation can be done in a quantum computer using a universal set of quantum gates.
This chapter points out the limitations in simulating quantum systems in classical computers. It describes the various stages of quantum simulation using quantum computers and relevant quantum circuits. Quantum simulators are classified into two types: analog quantum simulators and digital quantum simulators. In an analog quantum simulator the physical properties of a targeted model are reproduced on a physical set-up under externally controlled conditions. Quantum simulation will be considered successful only if the initial state preparation, the implementation of the time evolution and the measurements of observables are realized using only polynomial resources. The chapter discusses the quantum circuits for measurement of the dynamical correlation functions and the spectrum of the Hermitian operators. Computing the time evolution operator is equivalent to the task of implementing a unitary matrix and any unitary transformation can be done in a quantum computer using a universal set of quantum gates.
Quantum circuits If the state of a quantum system is a kind of information, then the dynamics of that system is a kind of information processing. This is the basic idea behind quantum computing , which seeks to exploit the physics of quantum systems to do useful information processing. In this chapter we will acquaint ourselves with a few of the ideas of quantum computing, using the idealized quantum circuit model. Then we will turn our attention to an actual quantum process, nuclear magnetic resonance , that can be understood as a realization of quantum information processing. In a quantum circuit, we have a set of n qubit systems whose dynamical evolution is completely under our control. We represent the evolution as a sequence of unitary operators, each of which acts on one or more of the qubits. Graphically, we represent the qubits as a set of horizontal lines, and the various stages of their evolution as a series of boxes or other symbols showing the structure of the sequence of operations. In such a diagram, time runs from left to right (see Fig. 18.1.). The n qubits form a kind of “computer,” and its overall evolution amounts to a “computation.” The key idea is that very complicated unitary operations on the n qubits can be built up step-by-step from many simple operations on one, two or a few qubits.
Quantum mechanics has proven its position as the primary theory for accurately describing microscopic domains. Numerous fields in physics utilize quantum mechanics to great effects, with one prominent application being quantum computing. Quantum computing extends the field of quantum mechanics to information theory, striving to perform computational tasks utilizing quantum phenomena with notably greater efficiency compared to classical alternatives. In this thesis, I will go through the basics required in the field of quantum computing. To understand quantum computing demands adapting a new type of matrix mechanical view. This involves considering Hilbert spaces, which are linear vector space having a well defined inner product. Hilbert spaces contain state vectors describing quantum states, as well as operators for altering and measuring these states. The dynamics of quantum systems are governed by three quantum postulates. These specify Hilbert spaces and state vectors as the fundamental components of quantum systems, Hermitian operators as providing the physical manifestation of these systems, and finally, the Schrödinger's equation to describe the time evolution of such systems. These notions are required in producing quantum computing. Quantum computing uses qubits as the computational basis, analogous to classical bits but with the added capability of quantum superposition. Qubits are represented as two-state vectors, and can be physically realized with any quantum-behaving binary-valued system, like the electron spins. Qubits can be combined into higher dimensional states with the use of tensor products, causing the system's Hilbert space to expand as 2^n with n being the number of qubits, allowing for exponential computational speedup. Computational logic can be constructed by acting on the qubits using unitary operators, known as quantum gates. The combinations of qubits and quantum gates can be visualized as quantum circuits, depicted using specialized circuit diagram notations. An essential part of quantum computing, and the phenomena enabling exponential computational speedup, is quantum entanglement. Entanglement arises in interacting qubits, intertwining them so that the individual states become correlated. Entanglement can be characterized by being unable to reduce a combined state into its pure constituents. The amount of entanglement in a system can be measured using von Neumann entropy, analogous to Shannon entropy in classical information theory. Entanglement serves various purposes in quantum computing and is a defining feature distinguishing it from the classical domain. In this thesis, the entanglement properties of a periodic four-qubit circuit with alternating quantum gates were simulated. The simulation was performed computationally, and measured entanglement entropy as a function of the amount of circuit periods. The results demonstrated different characteristics of entanglement depending on the parameters used for the quantum gates, with the entanglement either being periodic, non-occurring or chaotic in nature.
Quantum circuits—built from local unitary gates and local measurements—are a new playground for quantum many-body physics and a tractable setting to explore universal collective phenomena far from equilibrium. These models have shed light on longstanding questions about thermalization and chaos, and on the underlying universal dynamics of quantum information and entanglement. In addition, such models generate new sets of questions and give rise to phenomena with no traditional analog, such as dynamical phase transitions in quantum systems that are monitored by an external observer. Quantum circuit dynamics is also topical in view of experimental progress in building digital quantum simulators that allow control of precisely these ingredients. Randomness in the circuit elements allows a high level of theoretical control, with a key theme being mappings between real-time quantum dynamics and effective classical lattice models or dynamical processes. Many of the universal phenomena that can be identified in this tractable setting apply to much wider classes of more structured many-body dynamics.
Recent publications have introduced the concept of quantum-like (QL) bits, along with their associated QL-states and QL-gate operations, emerging from the dynamics of complex synchronized networks. The present work builds on this framework to present a simulation of quantum dynamics in spin systems. We focus on the transverse field Ising model, a widely studied Hamiltonian for interacting spin systems. We show how circuits, built using the QL analog of quantum gates, can be used to prepare an arbitrary initial wavepacket and simulate the time evolution of the resulting state for extracting physical observables such as magnetization. Beyond the statevector dynamics, we introduce a set of metrics that characterize how QL-circuit operations transform the underlying graph connections. These metrics quantify the microscopic coupling, phase coherence, and effective inter-block coupling generated by the circuit. These graph-level quantities provide constraints and design targets for potential physical implementations, indicating where interconnection resources, phase control, and effective coupling strength must be engineered in a hardware realization. Overall, this study demonstrates how classical resources can be harnessed to simulate quantum systems and we discuss the advantages and limitations.
In quantum computers a unitary transformation is applied to a given initial state of a set of qubits through several quantum gates. Tests were conducted using Grover's algorithm and implemented in a variety of different ways on the publically available IBM quantum computers. Essentially, quantum computers manipulate quantum states instead of classical bits. Quantum information can be represented by the elementary units called quantum bits abbreviated as qubits or qbits. This chapter discusses the fundamental properties of quantum systems that are relevant to information processing. Quantum computation needs something much more than that quantum parallelism to be useful. This is achieved in Deutsch's algorithm which combines quantum parallelism with interference. One can simulate a classical circuit using a quantum circuit. That is, it is possible to perform classical computations on quantum computers. But the advantage of quantum computing is that powerful functions may be computed by making use of quantum parallelism, the fundamental feature of many quantum algorithms.
Simulating quantum mechanics is known to be a difficult computational problem, especially when dealing with large systems. However, this difficulty may be overcome by using some controllable quantum system to study another less controllable or accessible quantum system, i.e., quantum simulation. Quantum simulation promises to have applications in the study of many problems in, e.g., condensed-matter physics, high-energy physics, atomic physics, quantum chemistry, and cosmology. Quantum simulation could be implemented using quantum computers, but also with simpler, analog devices that would require less control, and therefore, would be easier to construct. A number of quantum systems such as neutral atoms, ions, polar molecules, electrons in semiconductors, superconducting circuits, nuclear spins, and photons have been proposed as quantum simulators. This review outlines the main theoretical and experimental aspects of quantum simulation and emphasizes some of the challenges and promises of this fast-growing field.