Centre for Quantum Computer Technology

QUANTUM ALGORITHMS

Program Manager:
Prof Jason Twamley - Macquarie University

QUANTUM ALGORITHMS RESEARCHERS

Students
Mr Peter Brooke (PhD)
Mr Peter Morrison (MSc)
Mr Christopher Facer (Hon)

Staff
Prof Barry Sanders
Prof Igor Shparlinski
Dr James Cresser
A/Prof Bernard Mans
Dr Ron Steinfeld
Dr Manas Patra

OTHER COLLABORATORS
McGill - California Institute of Technology
Patrick Hayden - University Hong Kong
Bambi Hu - Baptist University
Peter Knight - Imperial College
Martin Plenio - Imperial College
Andrew Weily - Macquarie University
Karu Esselle - Macquarie University
Levente Horvath - Macquarie University
Tomas Tyc - Masaryk University
Sean Hallgren - NEC Laboratories America
Edo Waks - Stanford University
Eleni Diamanti - Stanford University
Yoshihisa Yamamoto - Stanford University
Howard Carmichael - University of Auckland
Shohini Ghose - University of Calga
Alex Russell - University of Connecticut
Kamleitner - University of Konstanz Ingo
James Clemens - University of Oregon
Stephen Barnett - University of Strathclyde
John Jeffers - University of Strathclyde
Peter Tan - University of Strathclyde
Stephen Bartlett - University of Sydney
Martin Rotteler - University of Waterloo
Michele Mosca - University of Waterloo
Raymond Laflamme - University of Waterloo
Casey Myers - University of Waterloo
Vladimir Buzek - Slovak Academy of Sciences
Mario Ziman - Slovak Academy of Sciences
Peter Stelmacovich - Slovak Academy of Sciences
Derek McHugh - Slovak Academy of Sciences
Christof Wunderlich - University of Siegen
Joerg Wrachtrup - University of Stuttgart
Fedor Jelezko - University of Stuttgart
Joe Fitzsimons - University of Oxford
Simon Benjamin - University of Oxford
Andrew Briggs - University of Oxford
Ian Walmsleymin - University of Oxford
Andrew Briggs - University of Oxford

PROGRAM DESCRIPTION
It is generally accepted now that the fabrication of a quantum processor of the scale required to break large cryptographic codes is far in the future. Much more emphasis is now being placed on determining much more modest, yet still revolutionary, uses for small scale quantum processors, where one might have control over 10-100 qubits. Such uses might range from relatively mundane quantum processing tasks such as in quantum repeaters or to more sophisticated tasks such as using a quantum computer to simulate a complex quantum system. As quantum processor fabrication technology advances, determining whether the processor is operating as expected has also become an intriguing question and protocols/algorithms will have to be developed to test ever more complex quantum processors. In the past year work in the quantum algorithms program has concentrated on a variety of goals, quantum random walks, the simulation of quantum systems, algorithms to estimate rational functions, through to studies of quantum noise and ways to protect against such noise, e.g. decoherence free subspaces. The program, in 2006, will greatly expand with new faculty and research fellow appointments at Macquarie University. As more uses for small-scale quantum processors emerge, the possibility of a truly quantum application driven technology will hopefully emerge.

Jason Twamley, in 2005, examined the popular topic of one dimensional quantum random walks, but from the novel viewpoint of inserting a spatially disordered region in the walk. Together with a former PhD student, Joseph Fitzsimons, Jason showed that the quantum wavepacket of the walker in the presence of spatial disorder experiences periods of super-ballistic spatial expansion, a most unusual, and potentially useful, result. Jason and Joe developed this study to discover a new type of spin-chain model of a quantum wire. Quantum wires may be useful in transporting quantum information around any future solid-state quantum processor. Jason & Joe’s quantum wire gives perfect qubit transmission, perfect multi-qubit transmission and with little extra, actually can perform quantum computation itself. Together with another former PhDstudent, Derek McHugh, Jason examined the operation of a qutrit, or base 3, quantum computer in trapped ions. Although not very typical, base 3 quantum computing may hold particular advantages over normal qubit quantum computation.

Figure 1 Schematic of quantum wire transport.

Barry Sanders and colleagues have found efficient algorithms for simulating sparse
Hamiltonians, which demonstrates that a quantum computer can simulate generic
Hamiltonian evolution with a cost in time nearly linear in evolution time and almost
constant cost in the number of qubits used. Barry has also recently developed a scheme
for generating large cross-phase modulation of two co-propagating slow pulses of light
in a Rubidium-87 gas. This demonstrates the possible creation of moderately large
optical Kerr nonlinearities which could be useful in many branches of quantum
information processing. Barry, with others, have developed a powerful new technique
for studying the potential improvement of efficiency for single photons and for singlerail
optical qubits using interferometry on multiple copies and postselection on all but
one channel. Unfortunately so far the results found suggest no improvement is possible
but a clear result either way remains to be found.

Figure 2 Schematic of method to generate moderate Kerr optical nonlinearities using double electrically induced transparency.

Igor Shparlinski has collaborated with leading international quantum computer
scientists to uncover two new results. The first problem is to determine a rational
function over a finite field using a quantum oracle which only returns several of the most significant bits of this function. Igor found a quantum algorithm which can deduce this rational function in polynomial time on a quantum computer while the corresponding classical problem is believed to be very hard. There may be potential uses in cryptography for Igor’s algorithm. Igor has also found a new construction for new types of symmetrically complete measurement operators (POVM), which may be useful in crafting new types of quantum algorithms. Both of these results have a strong underpinning in analytic number theory.

Understanding quantum noise and how it may impact the smooth operation of a quantum processor is an important area of study. James Cresser and colleagues have looked at two related problems. The first examines the age-old problem of quantum Brownian motion where the typical description leads to unphysical non- Markovian (history-dependent), dynamics. James has found a physically intuitive manner to derive a Markovian quantumdynamics for this system in terms of measurement theory and this work can help us understand when to treat the quantum noise in a Markovian manner. James and others have also developed this work to examine the general problem of describing continuous measurements of a system, and the relationship that this has to Lindblad master equations where frequent imperfectto- perfect measurements are made on a two level atom undergoing Rabi oscillations. Vastly different behaviours are seen depending on how perfect these frequent measurements are. Manas Patra, who has a background in both physics and computer science, has developed a formal language that is aimed at describing a quantum protocol or circuit in an abstract manner. He also has examined the optimality of measurement basis and has developed a measure to gauge how much information is lost when the measurement is sub-optimal.

In 2005, Peter Brooke, who is a PhD student with James Cresser and Barry Sanders
also found a number of interesting results. Firstly Peter and colleagues have derived
a formalism describing a collection of N co-located dipole-dipole interacting atoms,
and applied this to the collective emission properties of such a system. Secondly, Peter
has studied the influence of an off-diagonal unitary coupling on quantum information storage in decoherence free subsystems (DFS). He has shown that this coupling leads to a degradation of the DFS. Moreover he has found physical situations where this
detrimental effect is robust to variations in the symmetry condition. Unfortunately
this is bad news to many realistic DFS type schemes. Peter is continuing to apply these
ideas in a concrete condensed matter system.

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