This thesis explores the relationship between quantum computation and combinatorial structures, with the goal of improving our understanding of the complexity of quantum computation. We begin by studying the case when the complexity of combinatorial structures can be used to provide evidence for the hardness of classically simulating quantum computations. To this end, we show that the complexity of evaluating multiplicative-error approximations of Jones polynomials can be used to bound the classical complexity of simulating random quantum computations. We then proceed by studying the contrary case, that is, when do the combinatorial structures allow for an efficient classical simulation of quantum computations? We establish an efficient deterministic approximation algorithm for the Ising model partition function with complex parameters when the interactions and external fields are absolutely bounded close to zero. This provides an efficient classical algorithm for simulating a class of quantum computations with bounded interactions between the qubits.
In the second part of this thesis, we present some independent results on the efficient preparation of Fock states with a high number of photons from a resource of single photons. These Fock states are a fundamental resource in many quantum information protocols.
This thesis explores the relationship between quantum computation and combinatorial structures, with the goal of improving our understanding of the complexity of quantum computation. We begin by studying the case when the complexity of combinatorial structures can be used to provide evidence for the hardness of classically simulating quantum computations. To this end, we show that the complexity of evaluating multiplicative-error approximations of Jones polynomials can be used to bound the classical complexity of simulating random quantum computations. We then proceed by studying the contrary case, that is, when do the combinatorial structures allow for an efficient classical simulation of quantum computations? We establish an efficient deterministic approximation algorithm for the Ising model partition function with complex parameters when the interactions and external fields are absolutely bounded close to zero. This provides an efficient classical algorithm for simulating a class of quantum computations with bounded interactions between the qubits.
In the second part of this thesis, we present some independent results on the efficient preparation of Fock states with a high number of photons from a resource of single photons. These Fock states are a fundamental resource in many quantum information protocols.
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.
This thesis explores the interplay between structure and randomness in quantum computation, with the goal being to characterise the types of structure that give quantum computers an advantage over classical computation. The thesis begins by giving a necessary and sufficient condition for one notion of a quantum walk to be defined on a directed graph, and goes on to derive conditions on the structure of graphs that allow a quantum advantage in a non-local graph colouring game. A lower bound on entanglement-assisted quantum communication complexity based on information-theoretic ideas is given, and applied to the communication complexity of random functions. New lower bounds on the probability of success of quantum state discrimination are derived, and are applied to the problem of distinguishing random quantum states. This result is used to show a quantum advantage in almost all instances of a bounded-error single-query oracle identification problem. Lower bounds, and almost optimal algorithms, are given for two models of quantum search of partially ordered sets. This leads to the development of an optimal quantum algorithm to find the intersection of two sorted lists.
There is a natural relationship between Jones polynomials and quantum computation. We use this relationship to show that the complexity of evaluating relative-error approximations of Jones polynomials can be used to bound the classical complexity of approximately simulating random quantum computations. We prove that random quantum computations cannot be classically simulated up to a constant total variation distance, under the assumption that (1) the Polynomial Hierarchy does not collapse and (2) the average-case complexity of relative-error approximations of the Jones polynomial matches the worst-case complexity over a constant fraction of random links. Our results provide a straightforward relationship between the approximation of Jones polynomials and the complexity of random quantum computations.
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.
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.
Simulations play a crucial role in the investigation of complex quantum systems. In this thesis, locality structures of quantum systems are exploited to obtain complexity theoretic results with various physical implications. More specifically, rigorous mathematical tools are used and developed further, to investigate open quantum systems and thermal states. Moreover, important advances in photonic quantum simulations are discussed. For open quantum spin lattice systems new simulation schemes are provided. It is shown that Markovian dynamics can be simulated efficiently in the unitary circuit model, which can be seen as a dissipative Church-Turing type theorem. Moreover, Markovian dynamics is quasi-local and can be locally simulated on classical computers with a cost scaling polynomially in the system size. These results generalize standard tools from the investigation of Hamiltonian systems to open quantum systems. In particular, they provide a rigorous basis for their numerical simulation. However, also a major roadblock for making such simulations reliable is identified: Testing positivity of certain common approximations to mixed quantum states, called matrix product operators, is shown to be NP-hard in the system size and undecidable in the thermodynamic limit. Also more state space structures, originating from the spatial locality structure are discussed: Most states in state space cannot be generated efficiently with local Liouvillian dynamics and pure states generated in real- space renormalization schemes turn out to have local corrections in spatial dimensions larger than one. For thermal states on spin and fermionic lattice systems, a perturbation formula is provided and exponential clustering of correlations at high enough temperature is proven. This has various consequences: It leads to the extension of the concept of intensive temperature to interacting quantum systems, allows for efficient classical local simulations at high enough temperature, provides an upper bound on phase transition temperatures, and implies stability of thermal states against Hamiltonian perturbations. For photonic quantum simulations, sample complexity lower bounds for the verification of Boson-Sampling simulations are explained, which are applicable to a restricted setting. These bounds rely on a lower bound on the min-entropy of the output distribution of Boson-Sampling. This indicates that Boson-Sampling cannot be verified efficiently classically. Complementary to that, a reliable verification scheme for photonic state preparations is discussed, which uses single mode measurements as a simple quantum resource. The verification scheme is efficient for a large class of photonic simulations, including Boson-Sampling experiments with constantly many photons and state preparations necessary for measurement based quantum computing.