Phase Measurement &
Quantum Algorithms
Dominic Berry
IQC
Barry Sanders
Alex Lvovsky
University of Calgary
Andrew Childs
University of Waterloo
Jason Twamley
Alexei Gilchrist
Gavin Brennen
Ressa Said
Macquarie University
University of Waterloo
Howard Wiseman
Geoff Pryde
Brendon Higgins
Guoyong Xiang
Griffith University
Steve Bartlett
University of Sydney
Morgan Mitchell
ICFO
Tim Ralph
University of Queensland
Trevor Wheatley
Elanor Huntington
UNSW
Hidehiro Yonezawa
Daisuke Nakane
Hajime Arao
Akira Furusawa
University of Tokyo
Damian Pope
Perimeter Institute
Outline
1. Phase measurement
2. Anyon simulation
3. Photon processing
4. Quantum algorithms
5. Research plans
Phase measurement

Core work of the Centre for Quantum Dynamics.
Phase measurement
Distance measurement
Communication
Frequency and time
measurement
Phase measurement

Multipass interferometry

Nonadaptive interferometry

Multiphoton interferometry

Tracking a fluctuating phase

Loss resistant states

Magnetometry
Interferometry

N photons
 (t)
est

Simple inputs and measurements give
Standard Quantum Limit:
Interferometry

 (t)
N photons

est
More advanced inputs and measurements give
Heisenberg Limit:
Theoretical work with
Howard Wiseman showed
feedback can give this
result (PRL, 2000).
NOON state interferometry

input state
iN
e
N , 0  0, N
N , 0  0, N
est
Ambiguity problem due to
multiple fringes.
p( )
 /
B. C. Sanders, Phys. Rev. A 40, 2417 (1989).
Multipass interferometry

1 photon
 (t)
est
p( )
Similar ambiguity problem.
 /
Multipass interferometry

1 photon
 (t)
est
p( )
Resolving the ambiguity.
 /
Multipass interferometry

1 photon
 (t)
est
p( )
Resolving the ambiguity.
 /
Multipass interferometry

1 photon
 (t)
est
p( )
Resolving the ambiguity.
 /
Experimental results
B. L. Higgins, DWB, S. D. Bartlett, H. M. Wiseman & G. J. Pryde,
Nature 450, 393-396 (2007).
SQL
variance  N
M=1
M=6
number of resources, N
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 results
B. L. Higgins, DWB, S. D. Bartlett, M. W. Mitchell, H. M. Wiseman
& G. J. Pryde, New Journal of Physics 11, 073023 (2009).
standard deviation  N 1/2
SQL
hybrid
number of resources, N
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
standard deviation  N 1/2
G. Y. Xiang, B. L. Higgins, DWB, H. M. Wiseman & G. J. Pryde,
Nature Photonics 5, 43-47 (2011).
SQL
HL
photon number, N
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)
b)
c)
Signal and local
oscillator generation.
Adaptive phase
estimation.
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
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.
NOON
optimal loss tolerant
best from beam splitter
near-optimal states
coherent states
output

Magnetometry

Advances in nitrogenvacancy 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.

With low contrast,
nonadaptive measurements
are superior.
R. S. Said, DWB & J. Twamley,
Physical Review B (accepted
19 January, 2011).
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
DWB, M. Aguado, A. Gilchrist & G. K. Brennen,
New Journal of Physics 12, 053011 (2010).
Method to produce the required state:
7 
3 

 2 26  4
  arcsin 
 10
  arcsin 

pump


247 
single photon
Photon processing

1.
2.

Two major problems
for optical quantum
information:
inefficiency of photon
sources
photon loss
Can we recover from
these problems using
linear optics alone?
output
……
measurement
interferometer
……
……
input
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
……
measurement
interferometer
……
……
input
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).

New results show that, once we
have an appropriate definition of the
efficiency, linear optics cannot
increase the efficiency.
DWB & A. I. Lvovsky, Physical
Review Letters 105, 203601 (2010).
output
……
measurement
interferometer
……
……
input
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).

New results show that, once we
have an appropriate definition of the
efficiency, linear optics cannot
increase the efficiency.
DWB & A. I. Lvovsky, Physical
Review Letters 105, 203601 (2010).

Latest results indicate that we cannot
use some high-efficiency sources to
improve efficiency of other modes.
output
……
measurement
interferometer
……
……
input
DWB & A. I. Lvovsky,
arXiv:1010.6302 (2010).
Photon processing


A new way of quantifying vacuum in
modes.
Write annihilation operators for
modes as
aj  Bj Vj

Vj are vacuum annihilation operators.

We form matrix of commutators
output
……
measurement
interferometer
……
……
†
C jn  [ B j , B n ]
input

Non-vacuum component is
quantified by Ky Fan k-norm of C.
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.
 iH t /

e


This could be a quantum system – but a more general sparse
Hamiltonian can encode some other problem!
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.
 iH t /

e


This could be a quantum system – but a more general sparse
Hamiltonian can encode some other problem!
DWB, G. Ahokas, R. Cleve & B. C. Sanders,
Comm. Math. Phys. 270, 359 (2007).
NAND trees
A. M. Childs et al.,
Theory of
Computing 5, 119
(2009).
Simulation of Hamiltonians

The general problem is simulation of evolution under a
Hamiltonian.
 iH t /

e


This could be a quantum system – but a more general sparse
Hamiltonian can encode some other problem!
DWB, G. Ahokas, R. Cleve & B. C. Sanders,
Comm. Math. Phys. 270, 359 (2007).
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).
Simulation of Hamiltonians

The general problem is simulation of evolution under a
Hamiltonian.
 iH t /

e


This could be a quantum system – but a more general sparse
Hamiltonian can encode some other problem!
DWB, G. Ahokas, R. Cleve & B. C. Sanders,
Comm. Math. Phys. 270, 359 (2007).
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).
Quantum walks

An entirely new approach to simulating Hamiltonians.
wave

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.
DWB & A. M. Childs, arXiv:0910.4157 (2009).
Implementation of unitaries

A unitary is a general way of mapping a quantum state reversibly.

U 

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:
U
 0
H  †

U
0



Then the complexity of performing the unitary, in most cases,
scales as N .
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.
x  Ax  b
Using linear multistep methods, the problem can be encoded as
solution of a linear system:
Mx  b
The complexity then scales as


A t 
5/2
log N
Logarithmic in the dimension – an exponential speedup over
classical solution.
DWB, arXiv:1010.2745 (2010).
Research plans
Complementing and enhancing the
research activities of the school.
Centre for
Quantum
Dynamics
ARC Centre for
Quantum
Computation
and
Communication
Technology
Research plans
Complementing and enhancing the
research activities of the school.
ARC Centre for
Quantum
Computation
and
Communication
Technology
Centre for
Quantum
Dynamics
phase
measurement
optical
quantum
computing
quantum
algorithms
Research plans
Complementing and enhancing the
research activities of the school.
ARC Centre for
Quantum
Computation
and
Communication
Technology
Centre for
Quantum
Dynamics
phase
measurement
optical
quantum
computing
photon
processing
quantum
algorithms
anyon
simulation
Research plans
Phase 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 plans
Phase 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 plans
Optical quantum computing
Primary challenge is again to cope with photon loss.
output
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.
measurement
……
single photon
pump
interferometer
……
……
input
Research plans
Optical 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 plans
Quantum 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
phase
measurement
photon
processing
anyon
simulation
quantum
algorithms
optical
quantum
computing
Centre for
Quantum
Dynamics
ARC Centre for
Quantum
Computation
and
Communication
Technology
http://www.dominicberry.org/presentations/research.ppt