Wim van Dam

TCS Journal 2013 Journal Article

Quantum entanglement and the communication complexity of the inner product function

• Richard Cleve
• Wim van Dam
• Michael Nielsen
• Alain Tapp

We consider the communication complexity of the binary inner product function in a variation of the two-party scenario where the parties have an a priori supply of particles in an entangled quantum state. We prove linear lower bounds for both exact protocols, as well as for protocols that determine the answer with bounded-error probability. Our proofs employ a novel kind of “quantum” reduction from a quantum information theory problem to the problem of computing the inner product. The communication required for the former problem can then be bounded by an application of Holevo’s theorem. We also give a specific example of a probabilistic scenario where entanglement reduces the communication complexity of the inner product function by one bit.

SODA Conference 2007 Conference Paper

Quantum algorithm for a generalized hidden shift problem

• Andrew M. Childs
• Wim van Dam

FOCS Conference 2005 Conference Paper

From optimal measurement to efficient quantum algorithms for the hidden subgroup problem over semidirect product groups

• Dave Bacon
• Andrew M. Childs
• Wim van Dam

We approach the hidden subgroup problem by performing the so-called pretty good measurement on hidden subgroup states. For various groups that can be expressed as the semidirect product of an abelian group and a cyclic group, we show that the pretty good measurement is optimal and that its probability of success and unitary implementation are closely related to an average-case algebraic problem. By solving this problem, we find efficient quantum algorithms for a number of nonabelian hidden subgroup problems, including some for which no efficient algorithm was previously known: certain metacyclic groups as well as all groups of the form /spl Zopf//sub p/ /sup r/ /spl times/ /spl Zopf//sub p/ fixed r (including the Heisenberg group, r = 2). In particular our results show that entangled measurements across multiple copies of hidden subgroup states can be useful for efficiently solving the nonabelian HSP.

FOCS Conference 2004 Conference Paper

Adiabatic Quantum Computation is Equivalent to Standard Quantum Computation

• Dorit Aharonov
• Wim van Dam
• Julia Kempe
• Zeph Landau
• Seth Lloyd
• Oded Regev 0001

The model of adiabatic quantum computation has recently attracted attention in the physics and computer science communities, but its exact computational power has been unknown. We settle this question and describe an efficient adiabatic simulation of any given quantum algorithm. This implies that the adiabatic computation model and the standard quantum circuit model are polynomially equivalent. We also describe an extension of this result with implications to physical implementations of adiabatic computation. We believe that our result highlights the potential importance of the adiabatic computation model in the design of quantum algorithms and in their experimental realization.

SODA Conference 2003 Conference Paper

Quantum algorithms for some hidden shift problems

• Wim van Dam
• Sean Hallgren
• Lawrence Ip

FOCS Conference 2001 Conference Paper

How Powerful is Adiabatic Quantum Computation?

• Wim van Dam
• Michele Mosca
• Umesh V. Vazirani

The authors analyze the computational power and limitations of the recently proposed 'quantum adiabatic evolution algorithm'. Adiabatic quantum computation is a novel paradigm for the design of quantum algorithms; it is truly quantum in the sense that it can be used to speed up searching by a quadratic factor over any classical algorithm. On the question of whether this new paradigm may be used to efficiently solve NP-complete problems on a quantum computer, we show that the usual query complexity arguments cannot be used to rule out a polynomial time solution. On the other hand, we argue that the adiabatic approach may be thought of as a kind of 'quantum local search'. We design a family of minimization problems that is hard for such local search heuristics, and establish an exponential lower bound for the adiabatic algorithm for these problems. This provides insights into the limitations of this approach. It remains an open question whether adiabatic quantum computation can establish an exponential speed-up over traditional computing or if there exists a classical algorithm that can simulate the quantum adiabatic process efficiently.

STOC Conference 2000 Conference Paper

Self-testing of universal and fault-tolerant sets of quantum gates

• Wim van Dam
• Frédéric Magniez
• Michele Mosca
• Miklos Santha

FOCS Conference 1998 Conference Paper

Quantum Oracle Interrogation: Getting All Information for Almost Half the Price

• Wim van Dam

Consider a quantum computer in combination with a binary oracle of domain size N. It is shown how N/2+/spl radic/N calls to the oracle are sufficient to guess the whole content of the oracle (being an N bit string) with probability greater than 95%. This contrasts the power of classical computers which would require N calls to achieve the same task. From this result it follows that any function with the N bits of the oracle as input can be calculated using N/2+/spl radic/N queries if we allow a small probability of error. It is also shown that this error probability can be made arbitrary small by using N/2+O(/spl radic/N) oracle queries. In the second part of the article 'approximate interrogation' is considered. This is when only a certain fraction of the N oracle bits are requested. Also for this scenario does the quantum algorithm outperform the classical protocols. An example is given where a quantum procedure with N/10 queries returns a string of which 80% of the bits are correct. Any classical protocol would need 6N/10 queries to establish such a correctness ratio.