In this work, we introduce different types of quantum simulations according to different operator topologies on a Hilbert space, namely, uniform, strong, and weak quantum simulations. We show that they have the same computational power that the efficiently solvable problems are in bounded-error quantum polynomial time. For the weak simulation, we formalize a general weak quantum simulation problem and construct an algorithm which is valid for all instances. Also, we analyze the computational power of quantum simulations by proving the query lower bound for simulating a general quantum process.
In this work, we introduce different types of quantum simulations according to different operator topologies on a Hilbert space, namely, uniform, strong, and weak quantum simulations. We show that they have the same computational power that the efficiently solvable problems are in bounded-error quantum polynomial time. For the weak simulation, we formalize a general weak quantum simulation problem and construct an algorithm which is valid for all instances. Also, we analyze the computational power of quantum simulations by proving the query lower bound for simulating a general quantum process.
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.
Simulating quantum circuits using classical computers lets us analyse the inner workings of quantum algorithms. The most complete type of simulation, strong simulation, is believed to be generally inefficient. Nevertheless, several efficient strong simulation techniques are known for restricted families of quantum circuits and we develop an additional technique in this article. Further, we show that strong simulation algorithms perform another fundamental task: solving search problems. Efficient strong simulation techniques allow solutions to a class of search problems to be counted and found efficiently. This enhances the utility of strong simulation methods, known or yet to be discovered, and extends the class of search problems known to be efficiently simulable. Relating strong simulation to search problems also bounds the computational power of efficiently strongly simulable circuits; if they could solve all problems in P this would imply that all problems in NP and #P could be solved in polynomial time.
Quantum computation is by far the most powerful computational model allowed by the laws of physics. By carefully manipulating microscopic systems governed by quantum mechanics, one can efficiently solve computational problems that may be classically intractable; on the contrary, such speed-ups are rarely possible without the help of classical computation, since most quantum algorithms heavily rely on subroutines that are purely classical. A better understanding of the relationship between classical and quantum computation is indispensable, in particular in an era where the first quantum device exceeding classical computational power is within reach. In the first part of the thesis, we study some differences between classical and quantum computation. We first show that quantum cryptographic hashing is maximally resilient against classical leakage, a property beyond reach for any classical hash function. Next, we consider the limitation of strong (amplitude-wise) simulation of quantum computation. We prove an unconditional and explicit complexity lower bound for a category of simulations called monotone strong simulation and further prove conditional complexity lower bounds for general strong simulation techniques. Both results indicate that strong simulation is fundamentally unscalable. In the second part of the thesis, we propose classical algorithms that facilitate quantum computing. We propose a new classical algorithm for the synthesis of a quantum algorithm paradigm called quantum signal processing. Empirically, our algorithm demonstrates numerical stability and acceleration of more than one magnitude compared to state-of-the-art algorithms. Finally, we propose a randomized algorithm for transversally switching between arbitrary stabilizer quantum error-correcting codes. It has the property of preserving the code distance and thus might prove useful for designing fault-tolerant code-switching schemes.
Quantum computing has attracted a lot of attention in different research fields, such as mathematics, physics and computer science. Quantum algorithms can solve certain problems significantly faster than classical algorithms. There are many numerical problems, especially those arising from quantum systems, which are notoriously difficult to solve using classical computers, since the computational time required often scales exponentially with the size of the problem. However, quantum computers have the potential to solve these problems efficiently, which is also one of the founding ideas of the field of quantum computing. In this thesis, we explore five computational problems, designing innovative quantum algorithms and studying their computational complexity. First, we design an adiabatic quantum algorithm based on the Berry phases for the counting problem. For the running time, it is not as good as the optimal algorithm in the quantum circuit model, but better than the classical random algorithm. Moreover, since the Berry phase is a purely geometric feature, the result should be robust to decoherence and resilient to certain kinds of noise. Since the counting problem is the foundation of many other numerical problems, such as high-dimensional integration and path integration, our adiabatic algorithms can be directly generalized to these kinds of problems. In addition, we study the quantum PAC learning model, offering an improved lower bound on the query complexity. The lower bound is very close to the best lower bound on query complexity known for the classical PAC learning model. We also study the algorithms and the cost of simulating a system evolving under a given Hamiltonian. We consider high order splitting methods that are particularly applicable in quantum simulation and obtain bounds on the number of exponentials required. Moreover, we derive the optimal order of convergence given the required error bound. We compare our complexity estimates to previously known ones and show the resulting speedup. Furthermore, we consider randomized algorithms for Hamiltonian simulation. The evolution is simulated by a product of exponentials in a random sequence and random evolution times. Hence the final state of the system is approximated by a mixed quantum state. We provide a scheme to bound the error of the final quantum state in a randomized algorithm, and obtain randomized algorithms which have the same efficiency as certain deterministic algorithms but which are simpler to implement. We also apply Hamiltonian simulation in estimating the ground state energy of a multiparticle system, which is also known as the multivariate Sturm-Liouville eigenvalue problem. Since the cost of this problem grows exponentially with the number of particles using deterministic classical algorithms, it suffers from the curse of dimensionality. Quantum computers can vanquish the curse, and we exhibit a quantum algorithm whose total cost are linear in the number of particles.
Given oracle access to an unknown unitary U=eiH , the fractional query problem asks how many queries are required to implement a noninteger power Ut=eitH, 0<t<1, when the spectrum is separated from the branch cut by a gap δ. Quantum singular value transformation gives an upper bound of O(δ1logε1) queries for approximation error ε. We prove a matching lower bound for arbitrary query algorithms. Our argument reduces any N-query circuit to the approximation of eitθ by a trigonometric polynomial with degree bounded by O(N), together with Remez inequality. This allows us to establish the lower bound of Ωτ(δ1logε1). Consequently, the optimal query complexity for fractional query problem is Θτ(δ1logε1), showing that the known QSVT construction is asymptotically optimal. We also give an alternative lower bound proof based on constructing a linear functional that annihilates the approximant space, yielding a Ωτ(logε1) bound uniform to δ.
Abstract. A universal quantum simulator would enable efficient sim-ulation of quantum dynamics by implementing quantum-simulation al-gorithms on a quantum computer. Specifically the quantum simulator would efficiently generate qubit-string states that closely approximate physical states obtained from a broad class of dynamical evolutions. I provide an overview of theoretical research into universal quantum sim-ulators and the strategies for minimizing computational space and time costs. Applications to simulating many-body quantum simulation and solving linear equations are discussed