A quantum system that has interaction with external resources, such as probability distribution, dissipation, and noise, is referred to as an open quantum system. Not only do open quantum systems play a vital role in the field of quantum physics, but they are also fundamental objects in quantum information and quantum computing. In this thesis, we focus on computational problems related to open quantum systems. In particular, we study efficient constructions of open quantum systems and their algorithmic applications. \n \nA unitary 2-design is a quantum analogue of universal 2-hash functions. It is an example of open quantum systems in the sense that it is a probability distribution of unitaries. As unitary 2-designs inherit many properties of the Haar randomness on the unitary group, they have many applications in quantum information, such as benchmarking and decoupling. We study the structures of unitary 2-designs and present efficient methods for their constructions. \n \nThe continuous-time evolution of a closed quantum system can be described by the Schrödinger equation. A natural generalization of the Schrödinger equation to Markovian open quantum systems, in the sense of generating dynamical semigroups, is called the Lindblad equation. We show that it is impossible for a simple reductionist approach to simulate Lindblad evolution with gate complexity that has linear dependence in evolution time. Moreover, we use a novel variation of the "linear combination of unitaries" construction that pertains to quantum channels to achieve the desired linear dependence in evolution time and poly-logarithmic dependence in precision. \n \nOpen quantum systems can also be used as building blocks of quantum algorithms. We present a dissipative query model, which is based on the amplitude damping process. With this dissipative query model, we provide a quantum algorithm that performs a fixed-point quantum search while preserving the quadratic speedup against classical algorithms.
A quantum system that has interaction with external resources, such as probability distribution, dissipation, and noise, is referred to as an open quantum system. Not only do open quantum systems play a vital role in the field of quantum physics, but they are also fundamental objects in quantum information and quantum computing. In this thesis, we focus on computational problems related to open quantum systems. In particular, we study efficient constructions of open quantum systems and their algorithmic applications. \n \nA unitary 2-design is a quantum analogue of universal 2-hash functions. It is an example of open quantum systems in the sense that it is a probability distribution of unitaries. As unitary 2-designs inherit many properties of the Haar randomness on the unitary group, they have many applications in quantum information, such as benchmarking and decoupling. We study the structures of unitary 2-designs and present efficient methods for their constructions. \n \nThe continuous-time evolution of a closed quantum system can be described by the Schrödinger equation. A natural generalization of the Schrödinger equation to Markovian open quantum systems, in the sense of generating dynamical semigroups, is called the Lindblad equation. We show that it is impossible for a simple reductionist approach to simulate Lindblad evolution with gate complexity that has linear dependence in evolution time. Moreover, we use a novel variation of the "linear combination of unitaries" construction that pertains to quantum channels to achieve the desired linear dependence in evolution time and poly-logarithmic dependence in precision. \n \nOpen quantum systems can also be used as building blocks of quantum algorithms. We present a dissipative query model, which is based on the amplitude damping process. With this dissipative query model, we provide a quantum algorithm that performs a fixed-point quantum search while preserving the quadratic speedup against classical algorithms.
We consider the natural generalization of the Schrödinger equation to Markovian open system dynamics: the so-called the Lindblad equation. We give a quantum algorithm for simulating the evolution of an n-qubit system for time t within precision ε. If the Lindbladian consists of poly(n) operators that can each be expressed as a linear combination of poly(n) tensor products of Pauli operators then the gate cost of our algorithm is O(tpolylog(t/ε)poly(n)). We also obtain similar bounds for the cases where the Lindbladian consists of local operators, and where the Lindbladian consists of sparse operators. This is remarkable in light of evidence that we provide indicating that the above efficiency is impossible to attain by first expressing Lindblad evolution as Schrödinger evolution on a larger system and tracing out the ancillary system: the cost of such a \textit{reduction} incurs an efficiency overhead of O(t2/ε) even before the Hamiltonian evolution simulation begins. Instead, the approach of our algorithm is to use a novel variation of the "linear combinations of unitaries" construction that pertains to channels.
Closed quantum systems follow a unitary time evolution that can be simulated on quantum computers. By incorporating nonunitary effects via, e.g., measurements on ancilla qubits, these algorithms can be extended to open-system dynamics, such as Markovian processes described by the Lindblad master equation. In this paper, we analyze the convergence criteria and speed of a Markovian, purely dissipative quantum random walk on an unstructured classical search space. Notably, we show that certain jump operators make the quantum process replicate a classical one, while others yield differences between open quantum (OQRW) and classical random walks. We also clarify a previously observed quadratic speedup, demonstrating that OQRWs are no more efficient than classical search. Finally, we analyze a dissipative discrete-time ground-state preparation algorithm with a lower implementation cost. This allows us to interpolate between the dissipative and the unitary domain and thereby illustrate the important role of coherence for the quadratic speedup.
We consider the problem of constructing a ``universal set'' of Markovian processes, such that any Markovian open quantum system, described by a one-parameter semigroup of quantum channels, can be simulated through sequential simulations of processes from the universal set. In particular, for quantum systems of dimension d, we explicitly construct a universal set of semigroup generators, parametrized by ${d}^{2}\ensuremath{-}3$ continuous parameters, and prove that a necessary and sufficient condition for the dynamical simulation of a d-dimensional Markovian quantum system is the ability to implement (a) quantum channels from the semigroups generated by elements of the universal set of generators, and (b) unitary operations on the system. Furthermore, we provide an explicit algorithm for simulating the dynamics of a Markovian open quantum system using this universal set of generators, and show that it is efficient, with respect to this universal set, when the number of distinct Lindblad operators (representing physical dissipation processes) scales polynomially with respect to the number of subsystems.
Quantum computation appears to offer significant advantages over classical computation and this has generated a tremendous interest in the field. In this thesis we study the application of quantum computers to computational problems in science and engineering, and to combinatorial optimization problems. We outline the results below. Algorithms for scientific computing require modules, i.e., building blocks, implementing elementary numerical functions that have well-controlled numerical error, are uniformly scalable and reversible, and that can be implemented efficiently. We derive quantum algorithms and circuits for computing square roots, logarithms, and arbitrary fractional powers, and derive worst-case error and cost bounds. We describe a modular approach to quantum algorithm design as a first step towards numerical standards and mathematical libraries for quantum scientific computing. A fundamental but computationally hard problem in physics is to solve the time-independent Schrödinger equation. This is accomplished by computing the eigenvalues of the corresponding Hamiltonian operator. The eigenvalues describe the different energy levels of a system. The cost of classical deterministic algorithms computing these eigenvalues grows exponentially with the number of system degrees of freedom. The number of degrees of freedom is typically proportional to the number of particles in a physical system. We show an efficient quantum algorithm for approximating a constant number of low-order eigenvalues of a Hamiltonian using a perturbation approach. We apply this algorithm to a special case of the Schrödinger equation and show that our algorithm succeeds with high probability, and has cost that scales polynomially with the number of degrees of freedom and the reciprocal of the desired accuracy. This improves and extends earlier results on quantum algorithms for estimating the ground state energy. We consider the simulation of quantum mechanical systems on a quantum computer. We show a novel divide and conquer approach for Hamiltonian simulation. Using the Hamiltonian structure, we can obtain faster simulation algorithms. Considering a sum of Hamiltonians we split them into groups, simulate each group separately, and combine the partial results. Simulation is customized to take advantage of the properties of each group, and hence yield refined bounds to the overall simulation cost. We illustrate our results using the electronic structure problem of quantum chemistry, where we obtain significantly improved cost estimates under mild assumptions. We turn to combinatorial optimization problems. An important open question is whether quantum computers provide advantages for the approximation of classically hard combinatorial problems. A promising recently proposed approach of Farhi et al. is the Quantum Approximate Optimization Algorithm (QAOA). We study the application of QAOA to the Maximum Cut problem, and derive analytic performance bounds for the lowest circuit-depth realization, for both general and special classes of graphs. Along the way, we develop a general procedure for analyzing the performance of QAOA for other problems, and show an example demonstrating the difficulty of obtaining similar results for greater depth. We show a generalization of QAOA and its application to wider classes of combinatorial optimization problems, in particular, problems with feasibility constraints. We introduce the Quantum Alternating Operator Ansatz, which utilizes more general unitary operators than the original QAOA proposal. Our framework facilitates low-resource implementations for many applications which may be particularly suitable for early quantum computers. We specify design criteria, and develop a set of results and tools for mapping diverse problems to explicit quantum circuits. We derive constructions for several important prototypical problems including Maximum Independent Set, Graph Coloring, and the Traveling Salesman problem, and show appealing resource cost estimates for their implementations.
Quantum simulation, which is generically the task to employ quantum computers to simulate quantum physical models, has been one of the most significant motivations of quantum computing. Quantum dynamics, unitary or nonunitary Markovian dynamics driven by local interactions, has been proved to be efficiently simulatable on quantum computers. Extending the underlying models from unitary to nonunitary evolution, and from continuous-time to discrete-time evolution is essential not only for quantum simulation of more general processes, e.g., dissipative processes with evident non-Markovian effects, but also for developing alternative quantum computing models and algorithms. In this thesis, we explore quantum simulation problems mainly from the following three themes. First, we extend quantum simulation framework of Hamiltonian-driven evolution to quantum simulation of quantum channels, combined with the scheme of algorithmic simulation that accepts a promised simulation accuracy, hence algorithmic quantum channel simulation. Our simulation scheme contains a classical preprocessing part, i.e. a classical algorithm for quantum-circuit design, and a quantum part that is the execution of a quantum circuit for the quantum channel simulator. Second, we employ simulation algorithms for arbitrary quantum channels beyond the dilation method. We explore channel decomposition in terms of convex combination of smaller channels, known as generalized extreme channels, which is known as a nontrivial open problem. To attack this problem, we develop an optimization algorithm for approximate decomposition into a convex sum of generalized extreme channels. We provide an ansatz that proves to be able to yield arbitrary generalized extreme channels and allows a precise quantum circuit description. Furthermore, our numerical simulation has demonstrated the validity of our optimization algorithm for low-dimensional quantum channels. Third, by considering quantum simulation problems beyond the rough distinction between digital and analog simulations, and beyond the quantum-state generation problem, we define uniform, strong, and weak quantum simulations from the point view of operator topology. Besides strong simulation of quantum channels, we define a general weak quantum simulation problem, which simulates observable effects instead of the effects on state generation. Also we study the channel simulation problem in the quantum query model, and provide the query complexity by employing uniform quantum simulation method.
For the problem of simulating Lindblad evolution for time t to precision ε, Hamiltonian simulation provides an additive query lower bound, informally, Ω(t+polylog(1/ε)). However, the best previously known algorithms for general Lindblad simulation achieve a multiplicative upper bound, informally, O(tpolylog(1/ε)), in query complexity. It has remained open whether this multiplicative dependence is necessary. In this paper, we close the gap in query complexity by giving an algorithm with optimal additive dependence on evolution time and precision in the block-encoding model. Our approach uses the transducer framework to reduce the query cost of composing first-order approximations to the evolution channel, together with linear combinations of reuse circuits of different lengths to suppress catalyst-removal error. We further achieve nearly optimal gate complexity in evolution time and precision through history compression and an efficient implementation of the query-free part of the transducer using operation reordering and linear combinations of unitaries, while preserving the optimal query complexity.