We demonstrate a quantum error correction scheme that protects against
accidental measurement, using an encoding where the logical state of a single
qubit is encoded into two physical qubits using a non-deterministic photonic
CNOT gate. For the single qubit input states |0>, |1>, |0>+|1>, |0>-|1>,
|0>+i|1>, and |0>-i|1> our encoder produces the appropriate 2-qubit encoded
state with an average fidelity of 0.88(3) and the single qubit decoded states
have an average fidelity of 0.93(5) with the original state. We are able to
decode the 2-qubit state (up to a bit flip) by performing a measurement on one
of the qubits in the logical basis; we find that the 64 1-qubit decoded states
arising from 16 real and imaginary single qubit superposition inputs have an
average fidelity of 0.96(3).Comment: 4 pages, 4 figures, comments welcom

A. G. White

G. J. Pryde

J. L. O’Brien

M. A. Nielsen

T. C. Ralph

English

arXiv

O'Brien, JL

Pryde, GJ

White, AG

Ralph, TC

Griffith Research Online

Physical Review A

High-fidelity Z-measurement error encoding of optical qubits

We demonstrate a quantum error correction scheme that protects against accidental measurement, using a parity encoding where the logical state of a single qubit is encoded into two physical qubits using a nondeterministic photonic controlled-NOT gate. For the single qubit input states vertical bar 0 >, vertical bar 1 >, vertical bar 0 > +/- vertical bar 1 >, and vertical bar 0 > +/- i vertical bar 1 > our encoder produces the appropriate two-qubit encoded state with an average fidelity of 0.88 +/- 0.03 and the single qubit decoded states have an average fidelity of 0.93 +/- 0.05 with the original state. We are able to decode the two-qubit state (up to a bit flip) by performing a measurement on one of the qubits in the logical basis; we find that the 64 one-qubit decoded states arising from 16 real and imaginary single-qubit superposition inputs have an average fidelity of 0.96 +/- 0.03

O'Brien, J.L.

Pryde, G.J.

White, A.G.

Ralph, T.C.

University of Queensland eSpace

High-fidelity Z-measurement error encoding of optical qubits
J. L. O’Brien, G. J. Pryde, A. G. White, and T. C. Ralph
Centre for Quantum Computer Technology, Department of Physics, University of Queensland, 4072, Australia
sReceived 9 August 2004; published 9 June 2005d
We demonstrate a quantum error correction scheme that protects against accidental measurement, using a
parity encoding where the logical state of a single qubit is encoded into two physical qubits using a nondeter-
ministic photonic controlled-NOT gate. For the single qubit input states u0l, u1l, u0l± u1l, and u0l± iu1l our
encoder produces the appropriate two-qubit encoded state with an average fidelity of 0.88±0.03 and the single
qubit decoded states have an average fidelity of 0.93±0.05 with the original state. We are able to decode the
two-qubit state sup to a bit flipd by performing a measurement on one of the qubits in the logical basis; we find
that the 64 one-qubit decoded states arising from 16 real and imaginary single-qubit superposition inputs have
an average fidelity of 0.96±0.03.
DOI: 10.1103/PhysRevA.71.060303 PACS numberssd: 03.67.Lx, 03.67.Pp, 42.50.Dv
One of the greatest promises of quantum information sci-
ence is the exponential improvement in the computational
power offered by a quantum computer for certain tasks f1g.
Experimental progress has been made in NMR f2g, ion trap
f3,4g, cavity QED f5g, superconducting f6g, and spin f7g qu-
bits. A relatively recent proposal by Knill, Laflamme, and
Milburn sKLMd is linear optics quantum computing sLOQCd
f8g in which quantum information is encoded in single pho-
tons, and the nonlinear interaction required for two photon
gates is realized through conditional measurement f9–12g.
The critical challenge for all architectures is achieving fault
tolerance; this will require quantum error correction sQECd
f13,14g: a logical qubit uclL is encoded in a number of physi-
cal qubits such that joint measurements of the qubits can
extract information about errors without destroying the quan-
tum information. A five-qubit encoding against all one-qubit
errors has been demonstrated in NMR f15g. A two-qubit en-
coding, au00l+bu11l, has been demonstrated with polariza-
tion single photon qubits f16g.
A simple QEC code is the one introduced by KLM that
protects against a computational basis measurement—Z
measurement—of one of the qubits f8g
uclL = au0lL + bu1lL = asu00l + u11ld + bsu01l + u10ld .
s1d
This is a parity encoding: u0lL is represented by all even
parity combinations of the two qubits; u1lL by all the odd
parity combinations. If a Z measurement is made on either of
the physical qubits and the result “0” is obtained, then the
state collapses to an unencoded qubit, but the superposition
is preserved; if the result is “1” a bit-flipped version of the
unencoded qubit is the result. This generalizes straightfor-
wardly to an n-qubit code f17g: if a Z measurement occurs on
any of the qubits, the encoding is simply reduced to n−1
qubits. This type of QEC is critical for a scale up of LOQC
circuits: KLM showed that their nondeterministic, teleported
controlled-NOT sCNOTd gate f8g fails by performing a Z
measurement on one of the qubits sthis is also true for
equivalent gates f18,19gd. Thus by using this QEC technique
the qubits can be protected against gate failures and so the
effective success rate of gate operations can be boosted. A
similar principle underlies alternative LOQC schemes
f20,21g in which a quantum computation proceeds by gener-
ating a highly entangled state of many qubits—a cluster
state—and then performing only single qubit measurements
in a basis determined from the outcome of previous measure-
ments f22g. Like the n-qubit Z-error QEC code, these cluster
states are robust against accidental Z measurement. The en-
coding of Eq. s1d also forms the basic element in a redun-
dancy code, which can be used to correct for photon loss
errors f23g: uclLL=au0lLu0lL+bu1lLu1lL. For these reasons it
is important to show that qubit states can be encoded with
high fidelity and recovered with high fidelity after Z mea-
surements.
Here we report an experimental demonstration of QEC
using the two-qubit code of Eq. s1d, in which the encoded
state is prepared from an arbitrary input state, the error is
induced, and syndrome measured. The final bit flip correc-
tion is not made. A single qubit prepared in an arbitrary state
ucl=au0l+bu1l is input into the target mode of a nondeter-
ministic photonic CNOT gate. An ancilla qubit in the real
equal superposition u0l+ u1l is input into the control. We use
quantum state tomography to determine the resulting two-
qubit encoded state generated for the inputs ucl
= u0l , u1l , u0l± u1l, and u0l± iu1l sneglecting normalizationd,
and find an average fidelity of F=0.88±0.03 with uclL. For
the same six one-qubit inputs, the average fidelity of the
reconstructed one-qubit decoded states is F=0.93±0.05. Fi-
nally, we test the decoding sand encodingd by measuring one
or other of the qubits in the computational basis for 16 dif-
ferent real and imaginary superposition inputs and perform-
ing one-qubit quantum state tomography on the second qubit.
We find that the average fidelity of this second qubit with
the state ucl=au0l+bu1l or the bit-flipped version of it
uc8l=bu0l+au1l is F=0.96±0.03. These results demonstrate
that high fidelity Z-measurement QEC is possible for a
simple two-qubit code.
Figure 1sad shows schematically how the encoding of Eq.
s1d can be achieved using a CNOT gate and an ancilla qubit
prepared in the state u0l+ u1l. Figure 1sad also shows how a
projective measurement of one of the qubits in the logical
basis, plus a bit-flip X operation, conditional on the measure-
PHYSICAL REVIEW A 71, 060303sRd s2005d
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ment outcome, acts as a decoder. The CNOT gate used in our
demonstration is shown schematically in Fig. 1sbd. It oper-
ates nondeterministically and successful events are postse-
lected by coincidence measurements. For a detailed descrip-
tion of its operation see Ref. f10g. Pairs of energy degenerate
nonentangled photons of wavelength l=702.2 nm were gen-
erated in a nonlinear b-barium-borate sBBOd crystal through
spontaneous parametric down conversion of a l=351.1 nm,
P<900 mW pump beam. A coincidence window of 1 ns was
used and no correction for accidental counts was made.
These photons were sent through a polarizing beam splitter
sPBSd to prepare a highly pure horizontal polarization state.
Qubits are stored in the polarization state of these two pho-
tons using the assignment: horizontal uHl;u0l; and vertical
uVl;u1l. The control input is prepared in the equal real su-
perposition u0l+ u1l using a half wave plate sHWPd with its
optic axis at 22.5°. An arbitrary ucl was prepared using a
HWP and quarter wave plate sQWPd. The output of the cir-
cuit, nominally uclL, was analyzed using standard two-qubit
quantum state tomography f24g. The required two-qubit mea-
surements f24g were performed using a pair of analyzers
each consisting of a QWP and HWP followed by a PBS and
a single photon counting module. Each analyzer can perform
any one-qubit projective measurement.
We firstly confirm that this circuit produces the correct
two-qubit encoded state: Table I shows six one-qubit inputs
sthe two eigenstates and four equal superpositionsd and the
corresponding ideal encoded states. For these six one-qubit
input states we used our encoding circuit to produce a two-
qubit encoded state and performed two-qubit quantum state
tomography on the output. The real and imaginary parts of
the density matrices for the six two-qubit encoded states are
shown in Fig. 2. These experimentally measured encoded
states have an average fidelity of F=0.88±0.03 with uclL
swhere the error bar is the standard deviationd, demonstrating
the high fidelity of the encoding operation.
Next we test how well the encoding has worked by as-
suming the capability to perform perfect decoding and so
theoretically extract the one-qubit decoded states from the
two-qubit encoded states shown in Fig. 2. The one-qubit de-
coded state can be produced in four ways: by measuring
either the first or second qubit in the computational basis,
and getting the result “0” or “1.” In Fig. 3 we show all four
reconstructed one-qubit density matrices extracted from each
of the six encoded states in Fig. 2. We show the imaginary
components only for the imaginary superpositions since in
all other cases these components should be zero, and are
measured to be relatively small sthe average of imaginary
components is 0.04±0.04d. The average fidelity of these one-
FIG. 1. Encoding against Z-measurement error. sad A schematic
of a circuit for performing the encoding and decoding of the state
uclL. The circuit consisits of a CNOT gate with the control input set
to u0l+ u1l and the arbitrary, one-qubit state to be encoded ucl input
into the target. Note that decoding can be achieved by measuring
either of the encoded qubits. sbd Schematic of the experimental
CNOT gate shown in sad, as originally demonstrated in f10g.
TABLE I. One-qubit input states and the ideal corresponding
two-qubit encoded states.
1-qubit input 2-qubit encoded state Fig. 2
ucl=au0l+bu1l uCl=au00l+bu01l+bu10l+au11l
u0l u00l+ u11l 2sad
u1l u01l+ u10l 2sdd
u0l+ u1l u00l+ u01l+ u10l+ u11l 2sbd
u0l− u1l u00l− u01l− u10l+ u11l 2sed
u0l+ iu1l u00l+ iu01l+ iu10l+ u11l 2scd
u0l− iu1l u00l− iu01l− iu10l+ u11l 2sfd
FIG. 2. Two-qubit encoded states. The real and imaginary parts
of the density matrices are shown for the encoded states produced
for the one-qubit inputs given in Table I.
O’BRIEN et al. PHYSICAL REVIEW A 71, 060303sRd s2005d
RAPID COMMUNICATIONS
060303-2
qubit states with ucl or uc8l is F=0.93±0.05. Since these
states are reconstructed from the six two-qubit states, this
result demonstrates that given perfect decoding, the one-
qubit decoded states are more robust to imperfections in the
encoder than the two-qubit encoded states.
Finally we test the decoding circuit sas well as the encod-
ing circuitd by preparing the one-qubit input in the unequal
amplitude, real superpositions cossudu0l+sinsudu1l, and equal
amplitude, variable phase superpositions u0l+eis90°−2fdu1l, for
u, f=10°, 20°, 30°, 40°, 50°, 60°, 70°, and 80°. Together
with the results of Fig. 3 sgiving u, f=0° and 90°d, these
states map out two orthogonal great semicircles on the Bloch
sphere. For each of these inputs we reconstructed the one-
qubit density matrix directly for both measurement outcomes
on both qubits, i.e., four one-qubit density matrices for each
input state. From these density matrices we calculated the
fidelity with the ideal state ucl or uc8l. The results are shown
in Fig. 4. The average fidelity for all of these states sexclud-
ing u, f=0° and 90°d is 0.96±0.03, which is the same, to
within error, as for the reconstructed states shown in Fig. 3.
The behavior observed in Fig. 4 can be explained in terms
of the classical and nonclassical interference requirements of
the CNOT gate: The gate splits and recombines the control
and target polarization modes requiring two classical inter-
ferences, and nonclassically interferes the control uVl and
target uHl− uVl modes f10g. In practice, the visibility of each
of these interferences is subunity and thus contributes to er-
rors in the operation of the gate: The encoder works well for
u=f=45°, i.e., the input state u0l+ u1ls=uHl+ uVld since no
nonclassical interference is required, and only one classical
interference is required for the control. The encoder also
works well when qubit 1 sthe output of the control moded is
measured to be u0l since, again, only one classical interfer-
ence is required. Finally, the encoder works well for u close
to zero, which is expected from Fig. 2sad, which shows that
the u01lk01u population, the main contributor to errors in this
case, is very small.
This two-qubit Z-error encoding could be extended to a
n-qubit encoding using additional CNOT gates with single
photons prepared in the u0l+ u1l state as the control input,
and any of the already encoded qubits as the target. A very
similar technique can be used to build up a cluster state using
controlled-phase sCZd gates f20g. Note, however, that our
CNOT gate succeeds with the probability of 1 /9 f25g, but
can in principle be made deterministic and scalable f10g. In
order to build up the parity encoding or a cluster state in a
scalable way teleportation gates or similar techniques f21g
would be needed.
In conclusion, we have experimentally demonstrated a
high fidelity realisation of a two-qubit Z-measurement QEC
scheme sup to implementation of an X gated. For a represen-
tative set of input states the average fidelity of the one-qubit
decoded state is 0.96±0.03. Our scheme uses a nondetermin-
FIG. 3. One-qubit decoded states. A table of density matrices for
the six inputs and four decoding measurements listed in the figure.
The imaginary parts are shown for the u0l+ iu1l and u0l− iu1l inputs
only.
FIG. 4. One-qubit decoded state fidelities for the one-qubit input
states cossudu0l+sinsudu1l and u0l+eis90°−2fdu1l.
HIGH-FIDELITY Z-MEASUREMENT ERROR ENCODING… PHYSICAL REVIEW A 71, 060303sRd s2005d
RAPID COMMUNICATIONS
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istic CNOT gate operating on the polarization states of single
photon qubits, and is therefore a nondeterministic encoding
and will not be useful in its own right for a scalable quantum
computer. However, this does provide a proof of principle of
an encoding against a realistic error in linear optics quantum
computing. The technique can be extended to a larger en-
coded state or to the creation of a cluster state by using CZ
gates.
This work was supported by the Australian Research
Council, the NSA and ARDA under ARO Contract No.
DAAD 19-01-1-0651.
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High-Fidelity Z-Measurement Error Correction of Optical Qubits

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