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
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.
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 x&y; 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-1+1 queries]
Quantum mechanically: a single query suffices.
[Even for n-bit problem, since only yes/no outcome desired.]
Standard Deutsch-Jozsa Algorithm
Alice
0
1
Bob (oracle)
Alice
H
x
H
y y f(x)
x
f (0)  f (1)
H
0 1
2
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.
Photon number basis
Computational basis
1st qubit
1
1000
2
3
4
0100
01
0010
10
0001
11
00
2nd qubit
[Cf. Cerf, Adami, & Kwiat, PRA 57, R1477 (1998)]
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
NOT-1
X
CNOT-1
Hadamard-1
H
Four rails implementation
Quantum gate
00
01
10
11
00
01
10
11
NOT-2
00
01
10
11
00
01
10
11
CNOT-2
00
01
10
11
00+10
01+11
00-10
01-11
X
Hadamard-2
H
50/50 beam splitters
Four rails implementation
00
01
10
11
00
01
10
11
00
01
10
11
00
01
10
11
00
01
10
11
00+01
00-01
10+11
10-11
swap between two rails
Implementation of the oracle
The transformations introduced by the 4 possible functions or “oracles” can also be
implemented in this representation.
Constant oracle-00
00
f(0)=f(1)=0
00
Constant oracle-11
00
f(0)=f(1)=1
00
01
01
01
01
10
10
10
10
11
11
11
11
Balanced oracle-01
00
f(0)=0,f(1)=1
X
00
Balanced oracle-10
00
f(0)=1,f(1)=0
00
01
01
01
01
10
10
10
10
11
11
11
11
X
X
Error model and decoherence-free subspaces
Consider a source of dephasing which acts symmetrically
on states 01 and 10 (rails 2 and 3)…
00
01
00
eif 01
11 11
10 eif 10
e
2
i ( 1z  z
2 ) ( t ) / 2
But after oracle, only qubit 1 is needed for calculation.
Encode this logical qubit in either DFS: (00,11) or (01,10).
Modified Deutsch-Jozsa Quantum Circuit
0
H
x
1
H
y y f(x)
x
H
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).
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.
Interfering 1 with 4 and 2 with 3 is as effective as interfering
1 with 3 and 2 with 4 -- but insensitive to this decoherence model.
Experimental Setup
1
Random Noise
2
1
3
4
23
2
Preparation
Oracle
4
3
4
B
Swap
Phase Shifter
PBS
Detector
 / 2 Waveplate
Mirror
D
4
3
A
C
DJ without noise -- raw data
Original encoding
DFS Encoding
C
B
C
B
B
C
B
C
C
Constant function
B
Balanced function
Implementation of D-J without noise for both encoding
Original encoding
DFS Encoding
C
Normalized Intensity
1
0.95
0.9
0.85
B
C
B
B
C
B
C
0.8
0.75
0.7
0.65
0.6
0.55
Detectors A and C
Detectors B and D
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
0
10
20
30
40
Different Settings of oracle in time(s)
50
60
C
Constant function
B
Balanced function
Implementation of D-J in presence of noise
Original Encoding
Normalized intensity
DFS Encoding
C
1
0.95
0.9
0.85
0.8
0.75
0.7
0.65
0.6
0.55
0.5
0.45
0.4
0.35
0.3
0.25
0.2
0.15
0.1
0.05
0
B
C
B
B
C
B
C
Detectors A and C
Detectors B and D
0
10
20
30
40
Different settings of oracle in time(s)
50
60
C
Constant function
B
Balanced function
Coming Attractions:
Non-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.)
H-polarized photon
45o-polarized photon
Theory: how to distinguish nonorthogonal 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.
Experimental layout
(ancilla)
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.
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 nonorthogonal states 55% of the time, where standard quantum
measurements are limited to 33%. Also: "state filtering" or
discrimination of mixed states.