We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.
We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.
We study the complexity of approximating the smallest eigenvalue of a univariate Sturm-Liouville problem on a quantum computer. This general problem includes the special case of solving a one-dimensional Schroedinger equation with a given potential for the ground state energy. The Sturm-Liouville problem depends on a function q, which, in the case of the Schroedinger equation, can be identified with the potential function V. Recently Papageorgiou and Wozniakowski proved that quantum computers achieve an exponential reduction in the number of queries over the number needed in the classical worst-case and randomized settings for smooth functions q. Their method uses the (discretized) unitary propagator and arbitrary powers of it as a query ("power queries"). They showed that the Sturm-Liouville equation can be solved with O(log(1/e)) power queries, while the number of queries in the worst-case and randomized settings on a classical computer is polynomial in 1/e. This proves that a quantum computer with power queries achieves an exponential reduction in the number of queries compared to a classical computer. In this paper we show that the number of queries in Papageorgiou's and Wozniakowski's algorithm is asymptotically optimal. In particular we prove a matching lower bound of log(1/e) power queries, therefore showing that log(1/e) power queries are sufficient and necessary. Our proof is based on a frequency analysis technique, which examines the probability distribution of the final state of a quantum algorithm and the dependence of its Fourier transform on the input.
We study the approximation of the smallest eigenvalue of a Sturm-Liouville problem in the classical and quantum settings. We consider a univariate Sturm-Liouville eigenvalue problem with a nonnegative function q from the class C2([0,1]) and study the minimal number $n(\e)$ of function evaluations or queries that are necessary to compute an $\e$-approximation of the smallest eigenvalue. We prove that $n(\e)=Θ(\e^{-1/2})$ in the (deterministic) worst case setting, and $n(\e)=Θ(\e^{-2/5})$ in the randomized setting. The quantum setting offers a polynomial speedup with {\it bit} queries and an exponential speedup with {\it power} queries. Bit queries are similar to the oracle calls used in Grover's algorithm appropriately extended to real valued functions. Power queries are used for a number of problems including phase estimation. They are obtained by considering the propagator of the discretized system at a number of different time moments. They allow us to use powers of the unitary matrix exp(21iM), where M is an n×n matrix obtained from the standard discretization of the Sturm-Liouville differential operator. The quantum implementation of power queries by a number of elementary quantum gates that is polylog in n is an open issue.
Our understanding of complex mechanical processes is by our inability to solve the equations that govern them except for simple cases. Numerical simulation of systems appears to be our best option to understand, design and improve systems. It turns out, however, that computational problems in mechanics are notoriously difficult to treat numerically. The computational time that is required often scales exponentially with the size of the problem.
One of the most radical approaches for treating problems was proposed by Feytiman in 1982 [46]: he suggested that mechanics itself showed a promising way to simulate physics. This idea, the so called computer, showed its potential convincingly in one important regime with the development of Shor's integer factorization algorithm which improves exponentially on the best known classical algorithm.
In this thesis we explore six different computational problems from mechanics, study their computational complexity and try to find ways to remedy them. In the first problem we investigate the reasons behind the improved performance of Shor's and similar algorithms. We show that the key part in Shor's algorithm, the phase estimation algorithm, achieves its good performance through the use of power queries and we give lower bounds for all phase estimation algorithms that use power queries that match the known upper bounds. Our research indicates that problems that allow the use of power queries will achieve similar exponential improvements over classical algorithms. We then apply our lower bound technique for power queries to the Sturm-Liouville eigenvalue problem and show matching lower bounds to the upper bounds of Papageorgiou and Wozniakowski [85]. It seems to be very difficult, though, to find nontrivial instances of the Sturm-Lionville problem for which power queries can be simulated efficiently.
A computer differs from a classical computer that uses randomness, because it allows negative probabilities that can cancel each other (destructive interference). Ideally we would like to transfer classical randomized algorithms to the computer and get speed improvements. One of the simplest classical randomized algorithm is the random walk and we study the behavior of the continuous-time random walk. We analyze this random walk in one dimension and give analytical formulas for its behavior that demonstrate its interference properties. Is interference or cancellation really the most important advantage that a computer has over a classical computer? To answer that question we study the class StociMA of stochastic quantum algorithms that only use classical gates, but are given a witness, i.e. an arbitrary state that can guide the algorithm in computing the correct answer, but should not be able to fool it. We show that there exists a complete problem for this class, which we call the stoquastic local Hamiltonian problem. In this problem we try to compute the lowest eigenvalue of a Hamiltonian with interactions that span only a fixed number of particles and all contribute negatively. With the help of this problem we prove that MA ⊆ StocIMA ⊆ SBP ∪ QMA. This shows that interference is one of the most important parts of computation.
The simulation of the evolution of a general system in time requires a computational time that is exponential in the dimension of the system. But maybe the problem that we ask for is too general and we can simulate special systems in polynomial time. In particular it would be interesting to study systems of limited energy, i.e. for which the state at starting time consists mainly out of components with small energy. We model this in the theory of weighted reproducing kernel Hilbert spaces with two different sets of weights: product weights and finite-order weights. We will show that the information cost of computing the evolution for starting states from these spaces is tractable, i.e. the cost does not grow exponentially with the dimension of the problem.
Finally we study a computational problem from lattice chromodynamics (QCD). In most popular algorithms that treat problems in QCD the (gauged) Dirac matrix has to be inverted numerous times. Since this matrix is large, sparse, and ill-conditioned, iterative approaches have to be used. Unfortunately a direct application of methods like conjugate gradient (CC) or minimal residual algorithms seem to give poor performance in practice. We study a newly proposed multigrid method, adaptive smoothed aggregation [29], that has promise to overcome these difficulties We show that while classical CG's convergence becomes worse as the matrix becomes almost singular, adaptive smoothed aggregation will still perform well.
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.
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.
We have seen in the previous chapters that quantum computers seem to be more powerful than classical computers for certain problems. There are limits on the power of quantum computers, however. Since a classical computer can simulate a quantum computer, a quantum computer can only compute the same set of functions that a classical computer can. The advantage of using a quantum computer is that the amount of resources needed by a quantum algorithm might be much less than what is needed by the best classical algorithm. In Section 9.1 we briefly define some classical and quantum complexity classes and give some relationships between them. Most of the interesting questions relating classical and quantum complexity classes remain open. For example, we do not yet know if a quantum computer is capable of efficiently solving an NP-complete problem (defined later). One can prove upper bounds on the difficulty of a problem by providing an algorithm that solves that problem, and proving that it will work within in a given running time. But how does one prove a lower bound on the computational complexity of a problem? For example, if we wish to find the product of two n-bit numbers, computing the answer requires outputting roughly 2n bits and that requires Ω(n) steps (in any computing model with finite-sized gates). The best-known upper bound for integer multiplication is O(n log n log log n) steps. It has proved extremely difficult to derive non-trivial lower bounds on the computational complexity of a problem. Most of the known non-trivial lower bounds are in the ‘black-box’ model (for both classical and quantum computing), where we only query the input via a ‘black-box’ of a specific form. We discuss the black-box model in more detail in Section 9.2. We then sketch several approaches for proving black-box lower bounds. The first technique has been called the ‘hybrid method’ and was used to prove that quantum searching requires Ω(√n) queries to succeed with constant probability. The second technique is called the ‘polynomial method’. We then describe a technique based on ‘block sensitivity’, and conclude with a technique known as the ‘adversary method’.