Ground-state quantum computers mimic quantum-mechanical time evolution within the amplitudes of a time-independent quantum state. We explore the principles that constrain this mimicking. A no-cloning argument is found to impose strong restrictions. It is shown, however, that there is flexibility that can be exploited using quantum teleportation methods to improve ground-state quantum computer design.
Ground-state quantum computers mimic quantum-mechanical time evolution within the amplitudes of a time-independent quantum state. We explore the principles that constrain this mimicking. A no-cloning argument is found to impose strong restrictions. It is shown, however, that there is flexibility that can be exploited using quantum teleportation methods to improve ground-state quantum computer design.
In this thesis, I investigate aspects of local Hamiltonians in quantum computing. First, I focus on the Adiabatic Quantum Computing model, based on evolution with a time dependent Hamiltonian. I show that to succeed using AQC, the Hamiltonian involved must have local structure, which leads to a result about eigenvalue gaps from information theory. I also improve results about simulating quantum circuits with AQC. Second, I look at classically simulating time evolution with local Hamiltonians and finding their ground state properties. I give a numerical method for finding the ground state of translationally invariant Hamiltonians on an infinite tree. This method is based on imaginary time evolution within the Matrix Product State ansatz, and uses a new method for bringing the state back to the ansatz after each imaginary time step. I then use it to investigate the phase transition in the transverse field Ising model on the Bethe lattice. Third, I focus on locally constrained quantum problems Local Hamiltonian and Quantum Satisfiability and prove several new results about their complexity. Finally, I define a Hamiltonian Quantum Cellular Automaton, a continuous-time model of computation which doesn't require control during the computation process, only preparation of product initial states. I construct two of these, showing that time evolution with a simple, local, translationally invariant and time-independent Hamiltonian can be used to simulate quantum circuits.
Quantum teleportation, a cornerstone of quantum information science, enables the faithful transfer of an unknown quantum state without physically transmitting the particle. This remarkable process is fundamentally underpinned by two profound quantum mechanical principles: Bell non-locality and the no-cloning theorem. Bell non-locality, stemming from shared quantum entanglement, provides the instantaneous correlation necessary for quantum state transfer, challenging classical notions of local realism. Simultaneously, the no-cloning theorem asserts the impossibility of creating an identical copy of an arbitrary unknown quantum state, ensuring quantum information is transferred uniquely and the original state is irreversibly consumed. This paper explores these foundational pillars, elucidating their theoretical origins, mathematical formulations, and critical, synergistic roles in enabling quantum teleportation. We demonstrate how their combined action allows for precise state reconstruction at the receiver's location while rigorously preserving causality. The discussion extends to their profound implications for quantum communication, quantum computing, and our fundamental understanding of quantum reality, highlighting how their intrinsic relationship defines a unique framework for quantum information integrity.
Abstract One of the most important properties of quantum information, and the one ultimately responsible for its cryptographic applications, is that it can not be copied. That statement, however, is not completely accurate. While the no-cloning theorem of quantum mechanics prevents quantum information from being copied in space, the reversibility of microscopic physics actually requires that the information be copied in time. In spacetime as a whole, therefore, quantum information is widely replicated but in a restricted fashion. We fully characterize which regions of spacetime can all hold the same quantum information. Because quantum information can be delocalized through quantum error correction and teleportation, it need not follow well-defined trajectories. Instead, replication of the information in any configuration of spacetime regions not leading to violations of causality or the no-cloning principle is allowed. To demonstrate this, we answer the operational question of exactly when the information can be summoned to a set of spacetime points, showing how to do so efficiently using a combination of teleportation and codeword-stabilized quantum codes. This provides a simple and complete description of where and when a qubit can be located in spacetime, revealing a remarkable variety of possibilities.
Quantum computers are constantly growing in their number of qubits, but continue to suffer from restrictions such as the limited pairs of qubits that may interact with each other. Thus far, this problem is addressed by mapping and moving qubits to suitable positions for the interaction (known as quantum circuit mapping). However, this movement requires additional gates to be incorporated into the circuit, whose number should be kept as small as possible since each gate increases the likelihood of errors and decoherence. State-of-the-art mapping methods utilize swapping and bridging to move the qubits along the static paths of the coupling map---solving this problem without exploiting all means the quantum domain has to offer. In this paper, we propose to additionally exploit quantum teleportation as a possible complementary method. Quantum teleportation conceptually allows to move the state of a qubit over arbitrary long distances with constant overhead---providing the potential of determining cheaper mappings. The potential is demonstrated by a case study on the IBM Q Tokyo architecture which already shows promising improvements. With the emergence of larger quantum computing architectures, quantum teleportation will become more effective in generating cheaper mappings.
A central problem in quantum computing is the finding of an unknown target state that encodes the solution of a certain computational task. To accomplish this goal, the evolution from a given initial state is performed with an associated total Hamiltonian, which is a time-dependent combination of two time-independent Hamiltonians: the problem-Hamiltonian, whose ground state is the unknown target, and a driving-Hamiltonian, whose ground state is the initial state. Here, we analyze this computational problem in the light of optimal control theory, considering each Hamiltonian modulated by an independent control function. For bounded controls, there is a minimum total evolution time beyond which the target state can be exactly prepared. We show that below this minimum time a possible optimal solution consists of both controls constantly tuned at their upper bound, provided the gradients of the associated control Hamiltonian with respect to the controls (the switching functions) are positive. We refer to this type of solution as the double-bang solution. We show that the double-bang solution is optimal for a teleportation protocol up to the minimum time. Additionally, we combine the double-bang solution and the adiabatic gate teleportation protocol to implement universal quantum computing. This approach to quantum computing is very appealing because of its simplicity and experimental feasibility. To corroborate our analytical results, we propose the use of a numerical quantum optimal control technique adapted to limit the amplitude of the controls, which converges to the double-bang solution when the final evolution time is shorter than the minimum time. We compare the fidelity of the teleported state obtained for the numerically optimized two-control functions with the usual one-control function scheme and with the quantum approximate optimization algorithm (QAOA). We find that the two-control approach has a better performance than the other approaches. Moreover, we investigate the energetic cost and the robustness against systematic errors in the teleportation protocol, considering different schemes.