Rapid state purification of a superconducting charge qubit

1
Rapid state purification of a superconducting
charge qubit
E. J. Griffith1, C. D. Hill1, J. F. Ralph1, K.
Jacobs2 and H. M. Wiseman3
1. The Department of Electrical Engineering and
Electronics, The University of Liverpool. United
Kingdom. 2. The Department of Physics and
Astronomy, Louisiana State University. United
States. 3. Centre for Quantum Computer
Technology, Griffith University. Australia.
Constraints
Introduction
The superconducting charge qubit is subject to
particular constraints To reduce the
possibility of thermal excitations between the
two states, the minimum energy separation
(EJ2pwJ) should be kept large. The non zero
junction energy means the constantly rotating
Bloch vector can not be stopped when the xy-plane
has been reached. The only control is through the
charge ng induced by the bias voltage Vin. This
corresponds to rotations about the z-axis. There
is a linear relationship between ng and wz,
however ng is limited to avoid accessing the
neighbouring charge state. Measurement of the
island charge corresponds to measuring the
z-axis, and this can not be changed. Measurement
of any other axis is not permitted.
Purification of a qubit by using weak measurement
1 can be a time consuming process when the
measurement strength is significantly
reduced. However it has been previously
discovered that it is possible to accelerate the
purification process by using quantum feedback
2,3. Where the maximum purification rate is
obtained by placing the Bloch vector on the plane
orthogonal to the measurement axis, prior to each
measurement. Unfortunately, the nature of the
superconducting charge qubit makes the task of
holding the Bloch vector within this plane very
difficult 4. Therefore we propose a feedback
protocol which yields near optimal performance,
by using only one control field with simple
p-pulses.
Figure 3 The minimum energy separation should
ideally be large 5. Unfortunately, this non
zero energy implies the Bloch vector is always in
motion and so can not be held on the required
xy-plane.
Purity
The largest change in the average (im)purity
occurs whenever the Bloch vector lies on the
plane orthogonal to the measurement axis, as
is maximised when
Impurity
... Bloch vector can not be stopped on xy-plane!
Where q is the angle between the Bloch vector and
the measurement axis.
Proposed protocol
Figure 5 Qubit allowed to purify naturally
the bias control is set to ng0.5 so the qubit
evolves naturally. Under the continual weak
measurement the Bloch vector grows in length as
it purifies. The constant rotations about the
x-axis causes the Bloch vector to take a spiral
path in the yz-plane. However,once the
z-measurement exceeds a threshold (ZLIMIT) ...
Figure 1 Ideal feedback to the plane orthogonal
to the measurement axis yields the fastest
purification rate. However the perfect controls
would be difficult to achieve in practice.
Figure 4 There are constant rotations about the
x-axis due to Josephson tunnelling. Only
limited control is available over the z-axis
rotations. In addition the z-axis is
the only measurable axis.
the feedback protocol specifies a z-rotational
frequency which tilts the plane of rotation. The
Bloch vector can now be rotated on to the x-axis
using a p pulse
Tilt angle, a
Z-freq, wz(ng)
Duration, t
Figure 2 Voltage controlled charge qubit,
creates sx and sz rotations. The junction CJ,
gate Cg and parasitic CP capacitances can be
expressed as an effective capacitance Cq. Values
can be found in 6
Figure 6 Returning the Bloch vector to the
x-axis
The qubit is a small superconducting island
coupled to a bulk super-conductor via a single
Josephson junction of fixed frequency EJ The
qubit is controlled using only a voltage bias Vin
which induces a charge ng 5
Figure 7 shows a single simulated run of the
proposed practical protocol as per Figure 6.
Rotating to the x-axis is of key importance as
it is where the effect of the x-rotations is
least. Once the pulse has finished, the Bloch
vector remains near the xy-plane for
longer. Hence figure 9, the improvement in the
average purification rate compares favourably
with the ideal optimal purification protocol.
On average, the process of continuous weak
measurement 1 increases the purity of the
system. Qubit charge measurements correspond to
measurements along the Bloch z-axis. The Bloch
vector is randomly drawn towards one of the two
z-axis sphere poles as the measurement progresses
and information is extracted.
Weak measurement
For measurement strength g
In the Bloch sphere representation
Figure 7, 8 Single simulation and timing
diagrams
Qubit model
1 L. Diosi, quant-ph/0505075 (2005). 2 K.
Jacobs, Phys. Rev. A 67, 030301(R)
(2003). 3 H. M. Wiseman and J. F. Ralph, New
J. Phys. 8, 90 (2006). 4 J. F. Ralph, E. J.
Griffith, C. D. Hill, and T. D. Clark, SPIE Vol.
6244, 624403 (2006). 5 Y. Makhlin, G. Schön,
and A. Shnirman, Rev. Mod. Phys. 73, 357
(2001). 6 Y. A. Pashkin et al, Nature 421, 823
(2003).
Figure 9 The improvement in the average
purification rate as a function of impurity. The
proposed method closely matches the optimal
protocol, despite using only a single non-ideal
control field and measurement of a single axis.
Summary of Results