Abstract: In order to use quantum error-correcting codes to improve the performance of a quantum computer, it is necessary to be able to perform operations fault-tolerantly on encoded states. I present a theory of fault-tolerant operations on stabilizer codes based on symmetries of the code stabilizer. This allows a straightforward determination of which operations can be performed fault-tolerantly on a given code. I demonstrate that fault-tolerant universal computation is possible for any stabilizer code. I discuss a number of examples in more detail, including the five-quantum-bit code.

Abstract: Using nuclear magnetic resonance techniques with a solution of chloroform molecules we implement Grover's search algorithm for a system with four states. By performing a tomographic reconstruction of the density matrix during the computation good agreement is seen between theory and experiment. This provides the first complete experimental demonstration of loading an initial state into a quantum computer, performing a computation requiring fewer steps than on a classical computer, and then reading out the final state.

Abstract: Quantum computers1,2,3,4,5 can in principle exploit quantum-mechanical effects to perform computations (such as factoring large numbers or searching an unsorted database) more rapidly than classical computers1,2,6,7,8. But noise, loss of coherence, and manufacturing problems make constructing large-scale quantum computers difficult9,10,11,12,13. Although ion traps and optical cavities offer promising experimental approaches14,15, no quantum algorithm has yet been implemented with these systems. Here we report the experimental realization of a quantum algorithm using a bulk nuclear magnetic resonance technique16,17,18, in which the nuclear spins act as ‘quantum bits’19. The nuclear spins are particularly suited to this role because of their natural isolation from the environment. Our simple quantum computer solves a purely mathematical problem in fewer steps than is possible classically, requiring fewer ‘function calls’ than a classical computer to determine the global properties of an unknown function.

Abstract: A quantum computer has a clear advantage over a classical computer for exhaustive search The quantum mechanical algorithm for exhaustive search was originally derived by using subtle properties of a particular quantum mechanical operation called the Walsh-Hadamard (W-H) transform This paper shows that this algorithm can be implemented by replacing the W-H transform by almost any quantum mechanical operation This leads to several new applications where it improves the number of steps by a square root It also broadens the scope for implementation since it demonstrates quantum mechanical algorithms that can adapt to available technology

Abstract: Recent research has demonstrated that quantum computers can solve certain types of problems substantially faster than the known classical algorithms. These problems include factoring integers and certain physics simulations. Practical quantum computation requires overcoming the problems of environmental noise and operational errors, problems which appear to be much more severe than in classical computation due to the inherent fragility of quantum superpositions involving many degrees of freedom. Here we show that arbitrarily accurate quantum computations are possible provided that the error per operation is below a threshold value. The result is obtained by combining quantum error–correction, fault–tolerant state recovery, fault–tolerant encoding of operations and concatenation. It holds under physically realistic assumptions on the errors.

Abstract: In order to design a quantum circuit that performs a desired quantum computation, it is necessary to find a decomposition of the unitary matrix that represents that computation in terms of a sequence of quantum gate operations. To date, such designs have either been found by hand or by exhaustive enumeration of all possible circuit topologies. In this paper we propose an automated approach to quantum circuit design using search heuristics based on principles abstracted from evolutionary genetics, i.e. using a genetic programming algorithm adapted specially for this problem. We demonstrate the method on the task of discovering quantum circuit designs for quantum teleportation. We show that to find a given known circuit design (one which was hand-crafted by a human), the method considers roughly an order of magnitude fewer designs than naive enumeration. In addition, the method finds novel circuit designs superior to those previously known.

Abstract: We present a class of fast quantum algorithms, based on Bernstein and Vazirani's parity problem, that retrieves the entire contents of a quantum database $Y$ in a single query. The class includes binary search problems and coin-weighing problems. We compare the efficiency of these quantum algorithms with the classical algorithms that are bounded by the classical information-theoretic bound. We show the connection between classical algorithms based on several compression codes and our quantum-mechanical method.

Abstract: Quantum computation uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts by taking advantage of quantum parallelism. The unique characteristics of quantum theory may also be used to create a quantum associative memory with a capacity exponential in the number of neurons. This paper covers necessary high-level quantum mechanical ideas and introduces a simple quantum associative memory. Furthermore, it provides discussion, empirical results and directions for future work.

Abstract: A Quantum Computer is a new type of computer which can efficiently solve complex problems such as prime factorization. A quantum computer threatens the security of public key encryption systems because these systems rely on the fact that prime factorization is computationally difficult. Errors limit the effectiveness of quantum computers. Because of the exponential nature of quantum com puters, simulating the effect of errors on them requires a vast amount of processing and memory resources. In this paper we describe a parallel simulator which accesses the feasibility of quantum computers. We also derive and validate an analytical model of execution time for the simulator, which shows that parallel quantum computer simulation is very scalable.

Abstract: We present and evaluate several optimization and implementation techniques for string sorting. In particular, we study a recently published radix sorting algorithm, Forward radixsort, that has a provably good worst-case behavior. Our experimental results indicate that radix sorting is considerably faster (often more than twice as fast) than comparison-based sorting methods. This is true even for small input sequences. We also show that it is possible to implement a radixsort with good worst-case running time without sacrificing average-case performance. Our implementations are competitive with the best previously published string sorting programs.

Abstract: We consider the problem of inserting a new item into an ordered list of N-1 items. The length of an algorithm is measured by the number of comparisons it makes between the new item and items already on the list. Classically, determining the insertion point requires log N comparisons. We show that, for N large, no quantum algorithm can reduce the number of comparisons below log N/(2 loglog N).

Abstract: This paper gives a simple proof of why a quantum computer, despite being in all possible states simultaneously, needs at least 0.707 sqrt(N) queries to retrieve a desired item from an unsorted list of items. The proof is refined to show that a quantum computer would need at least 0.785 sqrt(N) queries. The quantum search algorithm needs precisely this many queries.

Abstract: In this note we study the power of so called query-limited computers. We compare the strength of a classical computer that is allowed to ask two questions to an NP-oracle with the strength of a quantum computer that is allowed only one such query. It is shown that any decision problem that requires two parallel (non-adaptive) SAT-queries on a classical computer can also be solved exactly by a quantum computer using only one SAT-oracle call, where both computations have polynomial time-complexity. Such a simulation is generally believed to be impossible for a one-query classical computer. The reduction also does not hold if we replace the SAT-oracle by a general black-box. This result gives therefore an example of how a quantum computer is probably more powerful than a classical computer. It also highlights the potential differences between quantum complexity results for general oracles when compared to results for more structured tasks like the SAT-problem.

Abstract: Two algorithms for sorting n! numbers on an n-star interconnection network are described. Both algorithms are based on arranging the n! processors of the n-star in a virtual (n - 1)- dimensional array. The first algorithm runs in O(n 3 log n time. This performance matches that of the fastest previously known algorithm for the same problem. In addition to providing a new paradigm for sorting on the n-star, the proposed algorithm has the advantage of being considerably simpler to state while requiring no recursion in its formulation. Its idea is to sort the input by repeatedly sorting the contents of all rows in each dimension of the (n - 1>algorithm presented in this paper is more efficient. It runs in O(n 2) time and thus provides an asymptotic improvement over its predecessors. However, it is more elaborate as it uses an existing result for sorting optimally on an (n-1)-dimensional array.

Abstract: Under the assumption that a quantum computer can exactly copy quantum superpositions, we show that NP-complete problems can be solved probabilistically in polynomial time. We also propose two methods that could potentially allow to avoid the use of a quantum copymachine. Supported by the Academy of Finland under grant 14047. To be presented at MCU'98, March 1998, Metz, France. TUCS Research Group Theory Group: Mathematical Methods in Computer Science

Abstract: Widespread interest in quantum computation was sparked by an algorithm of P. Shor for factoring integers on a quantum computer. We investigate here a more recent quantum algorithm of L. Grover for searching a database. This algorithm demonstrates a proven speed-up against the best possible classical algorithm for the same task.

Abstract: Withdrawn by the author due to irreparable errors. We present a quantum algorithm that in the black-box model performs a search in an ordered list of N elements. Using 3/4 log N + O(1) queries, it achieves a success probability of at least 1/2, whereas classically, log N - O(1) queries are needed to obtain constant success probability. Moreover, our algorithm employs the Haar transform and thus differs substantially from Grover's search algorithm and from algorithms relying on the quantum Fourier transform.

Abstract: : The majority of work today relating to quantum computing has provided results that are almost incomprehensible to even the most advanced computer architect. This paper uses the recent research results of quantum mechanical logic as building blocks to derive a computer architecture based on quantum devices that is consistent with existing system architectures. The new and admittedly exotic nature of quantum computing notwithstanding, this work presents a quantum mechanical analog of a finite state machine which may be realized from a present day architectural framework. Existing architectures are finite state sequential machines with binary data representations, asynchronous combinatorial Boolean logic and synchronous memory. Synchronous combinatorial logic with a binary data representation and implementations of D, T and JK flip-flops, which are the primary forms of synchronous memory, are presented using quantum and near quantum devices. Even the issue of reliable operation of quantum devices is addressed by suggesting that redundancy and quantum majority logic could perform single error correction at key points within the quantum system. Quantum devices can be a step in the evolution of advanced computer architectures.

Abstract: Information is something that can be encoded in the state of a physical system, and a computation is a task that can be performed with a physically realizable device. Therefore, since the physical world is fundamentally quantum mechanical, the foundations of information theory and computer science should be sought in quantum physics. In fact, quantum information has weird properties that contrast sharply with the familiar properties of classical information. A quantum computer -- a new type of machine that exploits the quantum properties of infomation -- could perform certain typ$s of calculations far more efficiently than any foreseeable classical computer. To build a functional quantum computer will be an enormous technical challenge. New methods for quantum error correction are being developed that can help to prevent a quantum computer from crashing.

Abstract: We present a simple and general simulation technique that transforms any black-box quantum algorithm (a la Grover's database search algorithm) to a quantum communication protocol for a related problem, in a way that fully exploits the quantum parallelism. This allows us to obtain new positive and negative results. The positive results are novel quantum communication protocols that are built from nontrivial quantum algorithms via this simulation. These protocols, combined with (old and new) classical lower bounds, are shown to provide the first asymptotic separation results between the quantum and classical (probabilistic) two-party communication complexity models. In particular, we obtain a quadratic separation for the bounded-error model, and an exponential separation for the zero-error model. The negative results transform known quantum communication lower bounds to computational lower bounds in the black-box model. In particular, we show that the quadratic speed-up achieved by Grover for the OR function is impossible for the PARITY function or the MAJORITY function in the bounded-error model, nor is it possible for the OR function itself in the exact case. This dichotomy naturally suggests a study of bounded-depth predicates (i.e. those in the polynomial hierarchy) between OR and MAJORITY. We present black-box algorithms that achieve near quadratic speed up for all such predicates.

Abstract: This thesis presents a formal model for quantum protocols, that is, cryptographic protocols based on quantum mechanics. It also develops a definition of security (which includes notions such as privacy and correctness) for quantum protocols. First, existing classical (non-quantum) definitions of security are studied. There are essentially two approaches: one based on Turing Machines and simulation, the other using information theory. Both approaches are explored in detail and compared, then a particular variation of the first approach is chosen. Security is defined by introducing an interface (also called a simulator) which is capable of translating an attack on protocol 1 into an attack on protocol 2. The profile of a protocol is a stochastic variable, defined as the output of the participants and the view of the adversary who performs the attack. If an interface exists such that the profile of protocol 1 is indistinguishable from the profile of protocol 2 with the help of this interface, then protocol 1 is at least as secure as protocol 2. An absolute notion of security is obtained by comparing the protocol implementation with its 'ideal' equivalent. In this approach, and in classical cryptography in general, indistinguishability of probability distributions is a crucial tool. The meaning of this notion is explored in the context of density matrices, the mathematical formalism which is used to describe the state of a quantum system. Various measures of distinguishability of quantum states relevant from a cryptographic point of view are investigated: probability of error, trace norm (related to the Kolmogorov distance), overlap (related to the Bhattacharyya coefficient) and Shannon distinguishability (defined through the mutual information). It is shown that if we require that the distinguishability between two families of quantum states vanish exponentially fast, these four distinguishability measures are equivalent. Along the way upper and lower bounds for the Shannon distinguishability are obtained. This is a very useful result, because explicit computation of the Shannon distinguishability between two quantum states is often difficult (it requires solving a transcendental equation), whereas the other distinguishability measures, in particular the trace norm, are usually much easier to compute. In the last chapter the two previous topics are combined. A formal model for quantum protocols is formulated. It uses Quantum Turing Machines to represent the participants, who communicate through a quantum channel. The different nature of quantum information as opposed to classical information is explored in detail, including the implications this has for the definition. As in the classical case, we introduce an interface to compare the security of two protocols for the same task. Subsequently some consequences of the proposed definition are investigated. In particular, strong evidence is given that one classical technique often used in proofs of security for classical protocols, called rewinding, is impossible in the quantum setting.

Abstract: An interesting classical result due to Jackson allows polynomial-time learning of the function class DNF using membership queries Since in most practical learning situations access to a membership oracle is unrealistic, this paper explores the possibility that quantum computation might allow a learning algorithm for DNF that relies only on example queries A natural extension of Fourier-based learning into the quantum domain is presented The algorithm requires only an example oracle, and it runs in O(sqrt(2^n)) time, a result that appears to be classically impossible The algorithm is unique among quantum algorithms in that it does not assume a priori knowledge of a function and does not operate on a superposition that includes all possible states

Abstract: A quantum algorithm for a class of highly structured combinatorial search problems is introduced. This algorithm finds a solution in a single step, contrasting with the linear growth in the number of steps required by the best classical algorithms as the problem size increases, and the exponential growth required by classical and quantum methods that ignore the problem structure. In some cases, the algorithm can also guarantee that insoluble problems, in fact, have no solutions, unlike previously proposed quantum search algorithms.

Abstract: In 1982 Feynman1 observed that quantum-mechanical systems have an information-processing capability much greater than that of corresponding classical systems, and could thus potentially be used to implement a new type of powerful computer. Three years later Deutsch2 described a quantum-mechanical Turing machine, showing that quantum computers could indeed be constructed. Since then there has been extensive research in this field, but although the theory is fairly well understood, actually building a quantum computer has proved extremely difficult. Only two methods have been used to demonstrate quantum logic gates: ion traps3,4 and nuclear magnetic resonance (NMR)5,6. NMR quantum computers have recently been used to solve a simple quantum algorithm—the two-bit Deutsch problem7,8. Here we show experimentally that such a computer can be used to implement a non-trivial fast quantum search algorithm initially developed by Grover9,10, which can be conducted faster than a comparable search on a classical computer.

Abstract: Quantum computation uses microscopic quantum level effects to perform computational tasks and has produced results that in some cases are exponentially faster than their classical counterparts. Choosing the best weights for a neural network is a time consuming problem that makes the harnessing of this ‘quantum parallelism’ appealing. This paper briefly covers necessary high-level quantum theory and introduces a model for a quantum neuron.

Abstract: Computational devices may be supplied with external sources of information (oracles). Quantum oracles may transmit phase information which is available to a quantum computer but not a classical computer. One consequence of this observation is that there is an oracle which is of no assistance to a classical computer but which allows a quantum computer to solve undecidable problems. Thus useful relativized separations between quantum and classical complexity classes must exclude the transmission of phase information from oracle to computer.

Abstract: A framework is presented for the design and analysis of quantum mechanical algorithms, the sqrt(N) step quantum search algorithm is an immediate consequence of this framework. It leads to several other search-type applications - several examples are presented. Also, it leads to quantum mechanical algorithms for problems not immediately connected with search - two such algorithms are presented for estimating the mean and median of statistical distributions. Both algorithms require fewer steps than the fastest possible classical algorithms; also both are considerably simpler and faster than existing quantum mechanical algorithms for the respective problems.

Abstract: A redundancy in the existing Deutsch-Jozsa quantum algorithm is removed and a refined algorithm, which reduces the size of the register and simplifies the function evaluation, is proposed. The refined version allows a simpler analysis of the use of entanglement between the qubits in the algorithm and provides criteria for deciding when the Deutsch-Jozsa algorithm constitutes a meaningful test of quantum computation.