The description of the state of an n-bit quantum system requires 2n-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dissertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measurement. This also sheds light on the power of computing with a finite number of quantum bits—using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton. We then consider the complexity of solving certain problems in the quantum black-box model, an information theoretic model that has been a rich source of insights into the nature of quantum computation. We derive nearly optimal lower bounds for several problems in this model, including that of approximating the median. We also give new, optimal algorithms for approximate medians and other order statistics.
The description of the state of an n-bit quantum system requires 2n-1 complex numbers. This exponentially large information capacity of quantum states has been exploited in recent results showing both exponential speed-up, and exponential savings in communication costs in solving certain problems using quantum computers. In this dissertation, we establish limitations on the ways in which the exponentially many degrees of freedom hidden in quantum states may be accessed. More specifically, we give tight bounds for random access codes, which allow us to encode classical information using quantum bits such that only a small portion of the encoded information may be recovered via a measurement. This also sheds light on the power of computing with a finite number of quantum bits—using these techniques, we show an exponential size lower bound for quantum finite automata for a problem which can be solved on a linear size classical automaton. We then consider the complexity of solving certain problems in the quantum black-box model, an information theoretic model that has been a rich source of insights into the nature of quantum computation. We derive nearly optimal lower bounds for several problems in this model, including that of approximating the median. We also give new, optimal algorithms for approximate medians and other order statistics.
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’.
Quantum computation and information has become a central research area in theoretical computer science. It studies how information is encoded in nature according to the laws of quantum mechanics and what this means for its computational power. On one hand, the ability of a quantum system to be in a superposition of states provides a way to encode an exponential amount of information in a quantum state. However, this information is accessible only indirectly via measurements. The goal of this thesis is precisely to investigate the fundamental question of how information is encoded and processed in quantum mechanical systems. (1) We investigate their power relative to classical encodings in the model of one-way communication complexity and show that they can be exponentially more efficient, answering a long standing open question in the area of quantum communication complexity. (2) We show how the theory developed for the study of quantum information can be employed in order to answer questions about classical coding theory and complexity. We resolve a main open question about the efficiency of Locally Decodable Codes by reducing the problem to one about quantum codes and solving the latter using tools from quantum information theory. This is the first example where quantum techniques are essential in resolving a purely classical question. (3) We investigate the power of quantum encodings in the area of cryptography. We study certain cryptographic primitives, such as Private Information Retrieval, Symmetrically-Private Information Retrieval and Coin Flipping.
We consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts. On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings. Finally, using this lower bound, we prove an exponential lower bound on the size of 1-way quantum finite automata for a family of languages accepted by linear sized deterministic finite automata.
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 consider the possibility of encoding m classical bits into much fewer n quantum bits so that an arbitrary bit from the original m bits can be recovered with a good probability, and we show that non-trivial quantum encodings exist that have no classical counterparts.On the other hand, we show that quantum encodings cannot be much more succint as compared to classical encodings, and we provide a lower bound on such quantum encodings.Finally, using this lower bound, we prove an exponential lower bound an the size of l-way quantum linite automata for a family of languages accepted by linear sized deterministic linite automata.
We study the power of quantum proofs, or more precisely, the power of quantum Merlin-Arthur (QMA) protocols, in two well studied models of quantum computation: the black box model and the communication complexity model. Our main results are obtained for the communication complexity model. For this model, we identify a complete promise problem for QMA protocols, the linear sub-spaces distance problem. The problem is of geometrical nature: each player gets a linear subspace of R/sup m/ and considers the sphere of unit vectors in that subspace. Their goal is to output 1 if the distance between the two spheres is very small (say, smaller than 0.1 /spl middot/ /spl radic/2) and 0 if the distance is very large (say, larger than 0.9 /spl middot/ /spl radic/2). We show that: 1. The QMA communication complexity of the problem is O(logm). 2. The (classical) MA communication complexity of the problem is /spl Omega/(m/sup /spl epsi//) (for some /spl epsi/ > 0). 3. The (standard) quantum communication complexity of the problem is /spl Omega/(/spl radic/m). In particular, this gives an exponential separation between QMA communication complexity and MA communication complexity. For the black box model we give several observations. First, we observe that the block sensitivity method, as well as the polynomial method for proving lower bounds for the number of queries, can both be extended to QMA protocols. We use these methods to obtain lower bounds for the QMA black box complexity of functions. In particular, we obtain a tight lower bound of /spl Omega/(N) for the QMA black box complexity of a random function, and a tight lower bound of /spl Omega/(/spl radic/N) for the QMA black box query complexity of NOR(X/sub 1/,..., X/sub n/). In particular, this shows that any attempt to give short quantum proofs for the class of languages Co - NP have to go beyond black box arguments. We also observe that for any Boolean function G(X/sub 1/,..., X/sub n/), if for both G and 7minus;G there are QMA black box protocols that make at most T queries to the black box, then there is a classical deterministic black box protocol for G that makes 0(T/sup 6/) queries to the black box. In particular, this shows that in the black box model QMA /spl cap/ Co - QMA = P. On the positive side, we observe that any (total or partial) Boolean function G(X/sub 1/,..., X/sub n/) has a QMA black box protocol with proofs of length N that makes only 0(/spl radic/N) queries to the black box. Finally, we observe a very simple proof for the exponential separation (for promise problems) between QMA black box complexity and (classical) MA black box complexity (first obtained by Watrous).