Quantum algorithms in the presence of decoherence: optical experiments

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Quantum algorithms in the presence of
decoherence optical experiments
Masoud Mohseni, Jeff Lundeen, Kevin Resch
and Aephraim Steinberg Department of
Physics, University of Toronto
Friendly neighborhood theorists Daniel Lidar,
Sara Schneider,... Helpful summer student
Guillaume Foucaud
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Motivation
Photons are an ideal system for carrying quantum
info. (Nonscalable) linear-optics quantum
computation may prove essential as part of
quantum communications links. Efficient
(scalable) linear-optical quantum computation
is a very promising avenue of research, relying
on the same toolbox (and more). In any quantum
computation scheme, the smoky dragon is
decoherence and errors. Without error
correction, quantum computation would be
nothing but a pipe dream. We demonstrate how
decoherence-free subspaces (DFSs) may be
incorporated into a prototype optical quantum
algorithm.
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Prototype algorithm Deutsch's Problem (2-qbit
version)
An oracle takes as input a bit x, and calculates
an unknown one-bit function f(x). quantum
version inputs xy outputs x y ? f(x) Our
mission, should we decide to accept
it Determine, with as few queries as possible,
whether or not f(0) f(1). Classically must
measure both f(0) and f(1). For n-bit
extension, need at least 2n-11 queries
Quantum mechanically a single query
suffices. Even for n-bit problem, since only
yes/no outcome desired.
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DJ algorithm and 4-rail qubits
Standard Deutsch-Jozsa Algorithm
Bob (oracle)
Alice
Alice
H
x
x
H
y
H
Physical realization of qubits
We use a four-rail representation of our two
physical qubits and encode the logical states
00, 01, 10 and 11 by a photon traveling down one
of the four optical rails numbered 1, 2, 3 and 4,
respectively.
Cf. Cerf, Adami, Kwiat, PRA 57, R1477 (1998)
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Implementation of simple gates
It is easy to implement a universal set up of one
and two qubit operations in such a representation

Quantum gate
Four rails implementation
Quantum gate
Four rails implementation
swap between two rails
50/50 beam splitters
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Implementation of the oracle
The transformations introduced by the 4 possible
functions or oracles can also be implemented in
this representation.
Balanced oracle-01 f(0)0,f(1)1
Balanced oracle-10 f(0)1,f(1)0
00
00
00
00
01
01
01
01
X
X
10
10
10
10
11
11
11
11
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Error model and decoherence-free subspaces
Consider a source of dephasing which acts
symmetrically on states 01 and 10 (rails 2 and 3)
DFSs see Lidar, Chuang, Whaley, PRL 81, 2594
(1998) et cetera. Implementations see Kwiat et
al., Science 290,498 (2000) and Kielpinski et
al., Science 291, 1013 (2001).
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Schematic of DJ, some failures
Schematic diagram of D-J interferometer
1
2
3
4
Oracle
1
00
2
01
3
10
4
11
1
2
3
4
Click at either det. 1 or det. 2 (i.e., qubit 1
low) indicates a constant function each looks at
an interferometer comparing the two halves of the
oracle.
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DJ experimental setup
Experimental Setup
1
2
1
3
4
23
2
Preparation
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Oracle
3
4
B
Swap
D
4
Phase Shifter
3
C
A
PBS
Detector
Waveplate
Mirror
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DJ without noise -- raw data
Original encoding
DFS Encoding
C
B
C
C
C
B
B
B
11
DJ without noise -- results
Original encoding
DFS Encoding
C
B
C
C
C
B
B
B
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DJ with noise-- results
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Coming AttractionsNon-orthogonal State
Discrimination

• Non-orthogonal quantum states cannot be
  distinguished
• with certainty.
• This is one of the central features of quantum
  information
• which leads to secure (eavesdrop-proof)
  communications.
• Crucial element we must learn how to
  distinguish quantum
• states as well as possible -- and we must
  know how well
• a potential eavesdropper could do.

(work with J. Bergou et al.)
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Theory how to distinguish non-orthogonal states
optimally
Step 1 Repeat the letters "POVM" over and over.
Step 2 Ask Janos, Mark, and Yuqing for help.
The view from the laboratory A measurement of a
two-state system can only yield two possible
results. If the measurement isn't guaranteed to
succeed, there are three possible results (1),
(2), and ("I don't know"). Therefore, to
discriminate between two non-orth. states, we
need to use an expanded (3D or more) system. To
distinguish 3 states, we need 4D or more.
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A test case
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Experimental layout
(ancilla)
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Success!
"Definitely 3"
"Definitely 2"
"Definitely 1"
"I don't know"
The correct state was identified 55 of the
time-- Much better than the 33 maximum for
standard measurements.
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Summary

• We have demonstrated the utility of
  decoherence-free subspaces in a prototype
  linear-optical quantum algorithm.
• The introduction of localized turbulent airflow
  produced a type of collective optical
  dephasing, leading to large error rates.
• With the DFS encoding, the error rate in the
  presence of noise was reduced to 7, essentially
  its pre-noise value.
• We note that the choice of a DFS may be easier
  to motivate via consideration of the physical
  system than from purely theoretical (quantum
  circuit) considerations!
• More recent results successfully distinguish
  among 3 non-orthogonal states 55 of the time,
  where standard quantum measurements are limited
  to 33. Also "state filtering" or
  discrimination of mixed states.