phase measurement & quantum algorithms dominic berry iqc university of waterloo howard wiseman...
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Phase Measurement & Quantum Algorithms
Dominic BerryIQC University of Waterloo
Howard Wiseman Geoff PrydeBrendon HigginsGuoyong Xiang Griffith University
Steve Bartlett University of Sydney
Morgan Mitchell ICFO
Tim Ralph University of Queensland
Trevor WheatleyElanor Huntington UNSW
Hidehiro YonezawaDaisuke NakaneHajime AraoAkira Furusawa University of Tokyo
Damian Pope Perimeter Institute
Barry SandersAlex Lvovsky University of Calgary
Andrew Childs University of Waterloo
Jason TwamleyAlexei GilchristGavin BrennenRessa Said Macquarie University
Outline
1. Phase measurement
2. Anyon simulation
3. Photon processing
4. Quantum algorithms
5. Research plans
Core work of the Centre for Quantum Dynamics.
Phase measurement
Communication
Frequency and time measurement
Distance measurement
Phase measurement
Phase measurement
Multipass interferometry
Nonadaptive interferometry
Multiphoton interferometry
Tracking a fluctuating phase
Loss resistant states
Magnetometry
Interferometry
Simple inputs and measurements give Standard Quantum Limit:
est
N photons
(t )
Interferometry
N photons est
More advanced inputs and measurements give Heisenberg Limit:
(t )
Theoretical work with Howard Wiseman showed feedback can give this result (PRL, 2000).
NOON state interferometry
input state
,0 0,N N
est
/
p( )
B. C. Sanders, Phys. Rev. A 40, 2417 (1989).
Ambiguity problem due to multiple fringes.
,0 0,iNe N N
Multipass interferometry
Similar ambiguity problem.
1 photon (t)
est
/
p( )
Multipass interferometry
Resolving the ambiguity.
/
p( )
1 photon
est
(t)
Multipass interferometry
/
p( )
(t)1 photon
est
Resolving the ambiguity.
Multipass interferometry
/
p( )
1 photon
est
Resolving the ambiguity.
(t)
Experimental results
M = 6
SQL
M = 1
vari
ance
N
number of resources, N
theoretical limit
B. L. Higgins, DWB, S. D. Bartlett, H. M. Wiseman & G. J. Pryde, Nature 450, 393-396 (2007).
Nonadaptive interferometry
Previously it was expected that we can’t achieve the theoretical limit without adaptive measurements.
Not so! We can achieve the theoretical limit with just a sequence of nonadaptive measurements and multiple passes.
Not only this, we can prove that it is at the theoretical limit!
Experimental resultsB. L. Higgins, DWB, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman & G. J. Pryde, New Journal of Physics 11, 073023 (2009).
number of resources, N
SQL
theoretical limit
nonadaptive
hybrid
stan
dard
dev
iati
on
N 1
/2
Multiphoton interferometry
Multiphoton interferometry
Multiphoton interferometry
Multiphoton interferometry
Use three different states.
Determine a sequence of states for a given total photon number N such that the final variance is minimised.
Use feedback such that the expected variance after the next detection is minimised.
Adaptive estimation with entanglement
HL
SQL
photon number, N
stan
dard
dev
iati
on
N 1
/2
G. Y. Xiang, B. L. Higgins, DWB, H. M. Wiseman & G. J. Pryde, Nature Photonics 5, 43-47 (2011).
Tracking a fluctuating phase
( )I t
DWB & H. M. Wiseman, Phys. Rev. A 65, 043803 (2002).
DWB & H. M. Wiseman, Phys. Rev. A 73, 063824 (2006).
Tracking a fluctuating phase
a) Signal and local oscillator generation.
b) Adaptive phase estimation.
c) Dual homodyne phase estimation.
LO = local oscillator;RF = radio-frequency;EOM = electro-optic
modulator;WGM = waveguide modulator; LPF = low-pass filter;MCC = mode-cleaning cavity;AOM = acousto-optic
modulator.
Tracking a fluctuating phase
dual homodyne filtered
T. A. Wheatley, DWB, H. Yonezawa, D. Nakane, H. Arao,D. T. Pope, T. C. Ralph, H. M. Wiseman, A. Furusawa &
E. H. Huntington, Physical Review Letters 104, 093601 (2010).
Loss resistant states NOON states are
very sensitive to loss.
States with optimal loss resistance are difficult to produce.
I am working on simpler methods to produce near-optimal states.
output
NOON
best from beam splitter
optimal loss tolerant
near-optimal states
coherent states
Magnetometry Advances in nitrogen-
vacancy centres offer ability to map magnetic fields at nanoscale resolution.
With longer T2 times, the measurements have a similar problem with ambiguity.
We can apply methods from optical measurements to obtain improved magnetic field measurements. R. S. Said, DWB & J. Twamley,
Physical Review B (accepted 19 January, 2011).
With low contrast, nonadaptive measurements are superior.
Anyon simulation Recall bosons and fermions
give different signs when exchanged.
Anyons are have more complicated behaviour – they give a phase or a more general group action.
Anyons can provide a basis for quantum computing with excellent error tolerance.
Simulated anyons can be produced experimentally.
Anyon simulation
Anyons are on a two-dimensional array of spins.
“Electric charges” are shown as diamonds.
“Magnetic charges” are shown as squares.
Charges correspond to excitations in the ground state of a Hamiltonian.
We take the smallest plaquette with nontrivial behaviour.
Anyon simulation
Method to produce the required state:
pump
single photon
10arcsin
247
7 3arcsin
42 26
DWB, M. Aguado, A. Gilchrist & G. K. Brennen, New Journal of Physics 12, 053011 (2010).
Photon processing
Two major problems for optical quantum information:
1. inefficiency of photon sources
2. photon loss
Can we recover from these problems using linear optics alone?
output
input
interferometer
measurement
…… ……
……
Photon processing Early results showed that we could
increase the single photon probability:
DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,
031806(R) (2004).
output
input
interferometer
measurement
…… ……
……
Photon processing Early results showed that we could
increase the single photon probability:
New results show that, once we have an appropriate definition of the efficiency, linear optics cannot increase the efficiency.
DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,
031806(R) (2004).
DWB & A. I. Lvovsky, Physical Review Letters 105, 203601 (2010).
output
input
interferometer
measurement
…… ……
……
Photon processing Early results showed that we could
increase the single photon probability:
New results show that, once we have an appropriate definition of the efficiency, linear optics cannot increase the efficiency.
Latest results indicate that we cannot use some high-efficiency sources to improve efficiency of other modes.
DWB, S. Scheel, B. C. Sanders & P. L. Knight, Physical Review A 69,
031806(R) (2004).
DWB & A. I. Lvovsky, arXiv:1010.6302 (2010).
DWB & A. I. Lvovsky, Physical Review Letters 105, 203601 (2010).
output
input
interferometer
measurement
…… ……
……
Photon processing A new way of quantifying vacuum in
modes.
Write annihilation operators for modes as
Vj are vacuum annihilation operators.
We form matrix of commutators
Non-vacuum component is quantified by Ky Fan k-norm of C.
j j ja B V
†[ , ]jn j nC B B
output
input
interferometer
measurement
…… ……
……
Quantum algorithms
Simulation of Hamiltonians
Quantum walks
Implementation of unitaries
Solving linear differential equations
Simulation of Hamiltonians Quantum computers could give an
exponential speedup in the simulation of quantum physical systems.
This is the original reason why Feynman proposed the idea of quantum computers.
The state of the system is encoded into the quantum computer.
Simulation of Hamiltonians The general problem is simulation of evolution under a
Hamiltonian.
This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!
/iHte
DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).
Simulation of Hamiltonians The general problem is simulation of evolution under a
Hamiltonian.
This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!
/iHte
NAND trees
A. M. Childs et al., Theory of
Computing 5, 119 (2009).
DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).
Simulation of Hamiltonians The general problem is simulation of evolution under a
Hamiltonian.
This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!
/iHte
NAND trees
A. M. Childs et al., Theory of
Computing 5, 119 (2009).
Systems of linear equations
A. W. Harrow et al., Phys. Rev. Lett. 103,
150502 (2009).
DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).
Simulation of Hamiltonians The general problem is simulation of evolution under a
Hamiltonian.
This could be a quantum system – but a more general sparse Hamiltonian can encode some other problem!
/iHte
NAND trees
A. M. Childs et al., Theory of
Computing 5, 119 (2009).
Systems of linear equations
A. W. Harrow et al., Phys. Rev. Lett. 103,
150502 (2009).
Differential equations
DWB, arXiv:1010.2745
(2010).
DWB, G. Ahokas, R. Cleve & B. C. Sanders, Comm. Math. Phys. 270, 359 (2007).
Quantum walks An entirely new approach to simulating Hamiltonians.
Quantum walks turn out to be universal for quantum computing!
A special type of quantum walk, called a Szegedy quantum walk, produces evolution related to that under the Hamiltonian.
By using a range of tricks, we can use the Szegedy quantum walk to simulate Hamiltonians far more efficiently.
wave
DWB & A. M. Childs, arXiv:0910.4157 (2009).
Implementation of unitaries A unitary is a general way of mapping a quantum state reversibly.
For dimension N, it takes at least N 2 elementary operations to perform the unitary (counting argument).
Alternatively, we can consider an oracle that gives the matrix elements of the unitary.
We can encode implementation of the unitary as a Hamiltonian simulation problem:
Then the complexity of performing the unitary, in most cases, scales as .
U
N
†
0
0
UH
U
DWB & A. M. Childs, arXiv:0910.4157 (2009).
Linear differential equations Most applications of supercomputers are in the form of large
systems of differential equations. A previous algorithm for nonlinear differential equations was not
efficient – try linear differential equations.
Using linear multistep methods, the problem can be encoded as solution of a linear system:
The complexity then scales as
Logarithmic in the dimension – an exponential speedup over classical solution.
Mx b
x Ax b
5/ 2logA t N
DWB, arXiv:1010.2745 (2010).
Research plansComplementing and enhancing the
research activities of the school.
Centre for Quantum Dynamics
ARC Centre for Quantum
Computation and
Communication Technology
Research plansComplementing and enhancing the
research activities of the school.
Centre for Quantum Dynamics
ARC Centre for Quantum
Computation and
Communication Technology
phase measurement
quantum algorithms
optical quantum
computing
Research plansComplementing and enhancing the
research activities of the school.
Centre for Quantum Dynamics
ARC Centre for Quantum
Computation and
Communication Technology
phase measurement
photon processing
anyon simulation
quantum algorithms
optical quantum
computing
Research plansPhase measurement
Primary challenge is to cope with photon loss.
1. Collaborate with Geoff Pryde & Centre for Quantum Dynamics to achieve experimental demonstration of proposal for loss tolerant states.
2. Develop new proposals for schemes with larger numbers of photons.
output
Research plansPhase measurement
Other collaborations:
1. Measurements of a fluctuating phase. Collaboration with Howard Wiseman (Centre for Quantum Dynamics) and researchers at UNSW and University of Tokyo to achieve adaptive measurements of a fluctuating phase with a squeezed beam.
2. Magnetometry with NV centres. Collaboration with Wrachtrup group at Universität Stuttgart and Jason Twamley at Macquarie University.
( )I t
Research plansOptical quantum computing
Primary challenge is again to cope with photon loss.
1. Parity states – methods to create and analyse. Collaboration with Geoff Pryde & Centre for Quantum Dynamics.
2. Develop new methods of optical quantum computing using ideas from simulation of nonabelian anyons.
3. Use photon processing theory to analyse loss tolerance in optical quantum computing.
pump
single photon
output
input
interferometer
measurement
…… ……
……
Research plansOptical quantum computing
Secondary challenge is to increase scale.
1. Hyperentanglement – exploit multiple degrees of freedom for each photon.
2. Heralded entanglement – enables more efficient construction of photonic cluster states.
3. Methods to use entangled particles to produce cluster states more directly. Possible collaboration with Dave Kielpinski.
Research plansQuantum algorithms
Solution of differential equations is an extremely promising area, with many open problems:
1. Can quantum walks be used for solving differential equations?
2. What information can be efficiently extracted from the states produced by algorithms for solving differential equations?
3. Can the efficiency be improved by using the variable time amplitude amplification of Ambainis?
4. Can time-dependent linear differential equations be efficiently simulated?
5. What about partial differential equations?
6. Are nonlinear differential equations fundamentally difficult to solve?
Summary
phase measurement
photon processing
anyon simulation
quantum algorithms
Summary
phase measurement
photon processing
anyon simulation
quantum algorithms
optical quantum
computing
Summary
Centre for Quantum Dynamics
ARC Centre for Quantum
Computation and
Communication Technology
phase measurement
photon processing
anyon simulation
quantum algorithms
optical quantum
computing
http://www.dominicberry.org/presentations/research.ppt