Wim van Dam

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

Adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its computational power was unknown. We describe an efficient adiabatic simulation of any given quantum algorithm, which implies that the adiabatic computation model and the conventional quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows stating the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of sparse matrices.

Quantum algorithms for algebraic problems

Quantum computers can execute algorithms that dramatically outperform classical computation. As the best-known example, Shor discovered an efficient quantum algorithm for factoring integers, whereas factoring appears to be difficult for classical computers. Understanding what other computational problems can be solved significantly faster using quantum algorithms is one of the major challenges in the theory of quantum computation, and such algorithms motivate the formidable task of building a large-scale quantum computer. This article reviews the current state of quantum algorithms, focusing on algorithms with superpolynomial speedup over classical computation and, in particular, on problems with an algebraic flavor.

Implausible consequences of superstrong nonlocality

We look at the consequences of so-called ‘superstrong nonlocal correlations’, which are hypothetical violations of Bell/CHSH inequalities that are stronger than quantum mechanics allows while still preventing the possibility of instantaneous communication. It is shown that the existence of maximally superstrong correlated bits implies that all distributed computations can be performed with a trivial amount of communication, i.e. with one bit. If one believes that Nature does not allow such a computational ‘free lunch’, then this result gives a reason why superstrong correlation are indeed not possible.

Experimental implementation of an adiabatic quantum optimization algorithm.

We report the realization of a nuclear magnetic resonance computer with three quantum bits that simulates an adiabatic quantum optimization algorithm. Adiabatic quantum algorithms offer new insight into how quantum resources can be used to solve hard problems. This experiment uses a particularly well-suited three quantum bit molecule and was made possible by introducing a technique that encodes general instances of the given optimization problem into an easily applicable Hamiltonian. Our results indicate an optimal run time of the adiabatic algorithm that agrees well with the prediction of a simple decoherence model.

Universal entanglement transformations without communication

We show that in the presence of arbitrary catalysts, any pure bipartite entangled state can be converted into any other to unlimited accuracy without the use of any communication, quantum or classical.

Optimal measurements for the dihedral hidden subgroup problem.

We consider the dihedral hidden subgroup problem as the problem of distinguishing hidden subgroup states. We show that the optimal measurement for solving this problem is the so-called pretty good measurement. We then prove that the success probability of this measurement exhibits a sharp threshold as a function of the density nu=k/log N, where k is the number of copies of the hidden subgroup state and 2N is the order of the dihedral group. In particular, for nu 1 the optimal measurement identifies the hidden subgroup with a probability of order unity. Thus the dihedral group provides an example of a group G for which Omega(log|G|) hidden subgroup states are necessary to solve the hidden subgroup problem. We also consider the optimal measurement for determining a single bit of the answer, and show that it exhibits the same threshold. Finally, we consider implementing the optimal measurement by a quantum circuit, and thereby establish further connections between the dihedral hidden subgroup problem and average case subset sum problems. In particular, we show that an efficient quantum algorithm for a restricted version of the optimal measurement would imply an efficient quantum algorithm for the subset sum problem, and conversely, that the ability to quantum sample from subset sum solutions allows one to implement the optimal measurement.

Recent progress in quantum algorithms

What quantum algorithms outperform classical computation and how do they do it?

Adiabatic Quantum Computation Is Equivalent to Standard Quantum Computation

The model of adiabatic quantum computation is a relatively recent model of quantum computation that has attracted attention in the physics and computer science communities. We describe an efficient adiabatic simulation of any given quantum circuit. This implies that the adiabatic computation model and the standard circuit-based quantum computation model are polynomially equivalent. Our result can be extended to the physically realistic setting of particles arranged on a two-dimensional grid with nearest neighbor interactions. The equivalence between the models allows one to state the main open problems in quantum computation using well-studied mathematical objects such as eigenvectors and spectral gaps of Hamiltonians.

Quantum Kolmogorov Complexity

In this paper we give a definition for quantum Kolmogorov complexity. In the classical setting, the Kolmogorov complexity of a string is the length of the shortest program that can produce this string as its output. It is a measure of the amount of innate randomness (or information) contained in the string. We define the quantum Kolmogorov complexity of a qubit string as the length of the shortest quantum input to a universal quantum Turing machine that produces the initial qubit string with high fidelity. The definition of P. Vitanyi (2001, IEEE Trans. Inform. Theory47, 2464?2479) measures the amount of classical information, whereas we consider the amount of quantum information in a qubit string. We argue that our definition is a natural and accurate representation of the amount of quantum information contained in a quantum state. Recently, P. Gacs (2001, J. Phys. A: Mathematical and General34, 6859?6880) also proposed two measures of quantum algorithmic entropy which are based on the existence of a universal semidensity matrix. The latter definitions are related to Vitanyis and the one presented in this article, respectively.

Quantum Algorithms for Weighing Matrices and Quadratic Residues

Abstract. In this article we investigate how we can employ the structure of combinatorial objects like Hadamard matrices and weighing matrices to devise new quantum algorithms. We show how the properties of a weighing matrix can be used to construct a problem for which the quantum query complexity is significantly lower than the classical one. It is pointed out that this scheme captures both Bernstein and Vaziranis inner-product protocol, as well as Grovers search algorithm. In the second part of the article we consider Paleys construction of Hadamard matrices, which relies on the properties of quadratic characters over finite fields. We design a query problem that uses the Legendre symbol χ (which indicates if an element of a finite field Fq is a quadratic residue or not). It is shown how for a shifted Legendre function fs(i)=χ(i+s) , the unknown s ∈ Fq can be obtained exactly with only two quantum calls to fs . This is in sharp contrast with the observation that any classical, probabilistic procedure requires more than log q + log ((1-ɛ )/2) queries to solve the same problem.