Abstract: We address the question of how a quantum computer can be used to simulate experiments on quantum systems in thermal equilibrium. We present two approaches for the preparation of the equilibrium state on a quantum computer. For both approaches, we show that the output state of the algorithm, after long enough time, is the desired equilibrium. We present a numerical analysis of one of these approaches for small systems. We show how equilibrium (time-)correlation functions can be efficiently estimated on a quantum computer, given a preparation of the equilibrium state. The quantum algorithms that we present are hard to simulate on a classical computer. This indicates that they could provide an exponential speedup over what can be achieved with a classical device.

Abstract: These notes discuss the quantum algorithms we know of that can solve problems significantly faster than the corresponding classical algorithms. So far, we have only discovered a few techniques which can produce speed up versus classical algorithms. It is not clear yet whether the reason for this is that we do not have enough intuition to discover more techniques, or that there are only a few problems for which quantum computers can significantly speed up the solution. In the first section of these notes, I try to explain why the recent results about quantum computing have been so surprising. This section comes from a talk I have been giving for several years now, and discusses the history of quantum computing and its relation to the mathematical foundations of computer science. In Sections 2 and 3, I talk about the quantum computing model and its relationship to physics. These sections rely heavily on two of my papers (SIAM J. Comp. 26 (1997), 1484-1509; Doc. Math. Extra Vol. ICM I (1998), 467-486). Sections 4 and 5 illustrate the general technique of using quantum Fourier transforms to find periodicity. Section 4 contains an algorithm of Dan Simon showing that quantum computers are likely to be exponentially faster than classical computers for some problems. Section 5 discusses my factoring algorithm, which was inspired in part by Dan Simon's paper. In the final section, I discuss Lov Grover's search algorithm, which illustrates a different technique for speeding up classical algorithms. These techniques for constructing faster algorithms for classical problems on quantum computers are the only two significant ones which have been discovered so far.

Abstract: The quantum search algorithm can be looked at as a technique for synthesizing a particular kind of superposition---one whose amplitude is concentrated in a single basis state. This basis state is defined by a binary function $f(\overline{x})$ that is nonzero in this desired basis state and zero everywhere else. This paper extends the quantum search algorithm to an algorithm that can create an arbitrarily specified superposition on a space of size $N$ in $O(\sqrt{N})$ steps. The superposition is specified by a complex valued function $f(\overline{x})$ that specifies the desired amplitude of the system in basis state $\overline{x}$.

Abstract: We introduce the Shifted Legendre Symbol Problem and some variants along with efficient quantum algorithms to solve them. The problems and their algorithms are different from previous work on quantum computation in that they do not appear to fit into the framework of the Hidden Subgroup Problem. The classical complexity of the problem is unknown despite the various results on the irregularity of Legendre Sequences.

Abstract: After carrying out a protocol for quantum key agreement over a noisy quantum channel, the parties Alice and Bob must process the raw key in order to end up with identical keys about which the adversary has virtually no information. In principle, both classical and quantum protocols can be used for this processing. It is a natural question which type of protocols is more powerful. We show that the limits of tolerable noise are identical for classical and quantum protocols in many cases. More specifically, we prove that a quantum state between two parties is entangled if and only if the classical random variables resulting from optimal measurements provide some mutual classical information between the parties. In addition, we present evidence which strongly suggests that the potentials of classical and of quantum protocols are equal in every situation. An important consequence, in the purely classical regime, of such a correspondence would be the existence of a classical counterpart of so-called bound entanglement, namely bound information that cannot be used for generating a secret key by any protocol. This stands in sharp contrast to what was previously believed.

Abstract: I investigate whether it would technologically and economically make sense to build database search engines based on Grover's quantum search algorithm. The answer is not fully conclusive but in my judgement rather negative.

Abstract: Recent developments in quantum technology have shown that quantum computers can provide a dramatic advantage over classical computers for some algorithms. Since most problems of real interest for genetic algorithms (GAs) have a vast search space [Holland, 1975], it seems appropriate to consider how quantum parallelism can be applied to GAs. In this paper we present a simple quantum approach to genetic algorithms and analyze its benefits and drawbacks. This is significant because to date there are only a handful of quantum algorithms [Williams and Clearwater, 1997].

Abstract: Given two unsorted lists each of length N that have a single common entry, a quantum computer can find that matching element with a work factor of $O(N^{3/4}\log N)$ (measured in quantum memory accesses and accesses to each list) The amount of quantum memory required is $O(N^{1/2})$ The quantum algorithm that accomplishes this consists of an inner Grover search combined with a partial sort all sitting inside of an outer Grover search

Abstract: Quantum computers have been predicted to be exponentially faster than their classical counterparts for some computations, such as the factoring of large numbers. (See the article by Charles Bennett in PHYSICS TODAY, October 1995, page 24.) Whereas classical computers perform operations on information stored as classical bits, which can be in one of two discrete states, quantum computers perform operations on quantum bits, or “qubits,” which can be put into any superposition of two quantum states.

Abstract: We describe a quantum version of the classical probabilistic algorithms a la Rabin. The quantum probabilistic algorithm is fully unitary and reversible, and can be used as a subroutine in larger quantum computations. As an example, we describe a polynomial time algorithm for counting the number of primes smaller than a given integer.

Abstract: We compare classical and quantum query complexities of total Boolean functions. It is known that for worst-case complexity, the gap between quantum and classical can be at most polynomial [3]. We show that for average-case complexity under the uniform distribution, quantum algorithms can be exponentially faster than classical algorithms. Under non-uniform distributions the gap can even be super-exponential. We also prove some general bounds for average-case complexity and show that the average-case quantum complexity of MAJORITY under the uniform distribution is nearly quadratically better than the classical complexity.

Abstract: Accounting for resources is the central issue in computational efficiency. We point out physical constraints implicit in information readout that have been overlooked in classical computing. The basic particle-counting mode of read-out sets a lower bound on the resources needed to implement a quantum computer. As a consequence, computers based on classical waves are as efficient as those based on single quantum particles.

Abstract: The previously proposed Heisenberg-type relation $ E_c t_c >> \hbar {\cal C}$ for the energy used by a quantum computer, the total computation time and the logical ("classical") complexity of the problem is verified for the following examples of quantum computations: preparation of the input state, two Hamiltonian versions of the Grover's algorithm, a model of "quantum telephone directory", a quantum-optical device factorizing numbers and the Shor's algorithm.

Abstract: A theoretical model of a quantum device which can factorize any number N in two steps i.e. by preparing an input state and performing a measurement is discussed. The analysis reveals that the duration of state preparation and measurement is proportional to N while the energy consumption grows like log N. These results suggest the existence of Heisenberg-type relation putting limits on the efficiency of a quantum computer in terms of a total computation time, a total energy consumption and a classical complexity of the problem.

Abstract: In this paper, we show structures of efficient quantum algorithms on a quantum Turing machine. First, we denote a relationship between Simon's algorithm and Walsh transforms. Next, we also show that a quantum Turing machine can efficiently solve the Deutsch–Jozsa problem by using interference among configurations instead of observation. Moreover, based on these results, we are led to structures of more general quantum algorithms. Finally, we show a problem that can be efficiently solved by using this type of algorithm and a Walsh transform. © 2000 Scripta Technica, Electron Comm Jpn Pt 3, 83(8): 71–80, 2000

Abstract: We consider a quantum computational algorithm that can be used to determine (probabilistically) how close a given signal is to one of a set of previously observed signals stored in the state of a quantum neurocomputional machine. The realization of a new quantum algorithm for factorization of integers by Shor and its implication to cryptography has created a rapidly growing field of investigation. Although no physical realization of a quantum computer is available, a number of software systems simulating a quantum computation process exist. In light of the rapidly increasing power of desktop computers and their ability to carry out these simulations, it is worthwhile to investigate possible advantages as well as realizations of quantum algorithms in signal processing applications. The algorithm presented offers a glimpse of the potential of this approach. Neural networks (NN) provide a natural paradigm for parallel and distributed processing of a wide class of signals. Neural networks within the context of classical computation have been used for approximation and classification tasks with some success. We propose a model for quantum neurocomputation (QN) and explore some of its properties and potential applications to signal processing in an information theoretic context.

Abstract: Quantum information processing offers potentially great advantages over classical information processing, both for efficient algorithms1,2 and for secure communication3,4. Therefore, it is important to establish that scalable control of a large number of quantum bits (qubits) can be achieved in practice. There are a rapidly growing number of proposed device technologies5,6,7,8,9,10,11 for quantum information processing. Of these technologies, those exploiting nuclear magnetic resonance (NMR) have been the first to demonstrate non-trivial quantum algorithms with small numbers of qubits12,13,14,15,16. To compare different physical realizations of quantum information processors, it is necessary to establish benchmark experiments that are independent of the underlying physical system, and that demonstrate reliable and coherent control of a reasonable number of qubits. Here we report an experimental realization of an algorithmic benchmark using an NMR technique that involves coherent manipulation of seven qubits. Moreover, our experimental procedure can be used as a reliable and efficient method for creating a standard pseudopure state, the first step for implementing traditional quantum algorithms in liquid state NMR systems. The benchmark and the techniques can be adapted for use with other proposed quantum devices.

Abstract: Quantum robots are described as mobile quantum computers and ancillary systems that move in and interact with arbitrary environments. Their dynamics is given as tasks which consist of sequences of alternating computation and action phases. A task example is considered in which a quantum robot searches a space region to find the location of a system. The possibility that the search can be more efficient than a classical search is examined by considering use of Grover's Algorithm to process the search results. This is problematic for two reasons. One is the removal of entanglements generated by the (reversible) search process. The other is that (ignoring the entanglement problem), the search process in 2 dimensional space regions is no more efficient than a classical search. However quantum searches of higher dimensional space regions are more efficient than classical searches. Reasons why quantum robots are interesting independent of these results are briefly summarized.

Abstract: An alternative quantum algorithm for combinatorial search, adjusting amplitudes based on number of conflicts in search states, performs well, on average, for hard random satisfiability problems near a phase transition in search difficulty The algorithm exploits correlations among problem properties more effectively than some current heuristics, and improves on prior quantum algorithms that ignore these correlations

Abstract: Candidates for quantum computing which offer only restricted control, e.g., due to lack of access to individual qubits, are not useful for general purpose quantum computing. We present concrete proposals for the use of systems with such limitations as RISQ - reduced instruction set quantum computers and devices - for simulation of quantum dynamics, for multi-particle entanglement and squeezing of collective spin variables. These tasks are useful in their own right, and they also provide experimental probes for the functioning of quantum gates in pre-mature proto-types of quantum computers.

Abstract: We propose a new method for proving lower bounds on quantum query algorithms. Instead of a classical adversary that runs the algorithm with on input and then modifies the input, we use a quantum adversary that runs the algorithm with a superposition of inputs. If the algorithm works correctly, its state becomes entangled with the superposition over inputs. We bound the number of queries needed to achieve a sufficient entanglement and this implies a lower bound on the number of queries for the computation. Using this method, we prove two new Ω(√N) lower bounds on computing AND of ORs and inverting a permutation and also provide more uniform proofs for several known lower bounds which have been previously proven via a variety of different techniques.

Abstract: A quantum algorithm for an oracle problem can be understood as a quantum strategy for a player in a two-player zero-sum game in which the other player is constrained to play classically. I formalize this correspondence and give examples of games (and hence oracle problems) for which the quantum player can do better than would be possible classically. The most remarkable example is the Bernstein-Vazirani quantum search algorithm which I show creates no entanglement at any timestep.