Rapid state purification in a superconducting charge qubit
E J Griffith1, C D Hill1, J F Ralph1, K Jacobs2 and H M Wiseman3
1. Department of Electrical Engineering and Electronics, The University of Liverpool,
Brownlow Hill, Liverpool, L69 3GJ, United Kingdom.
2. Department of Physics and Astronomy, Louisiana State University, Nicholson Hall,
Tower Drive, Baton Rouge, LA 70803, USA.
3. Centre for Quantum Computer Technology, Griffith University, Brisbane,
Queensland 4111, Australia.
e.j.griffith@liverpool.ac.uk
Abstract. We consider a technique for implementing a rapid state purification scheme, within
the constraints present in a superconducting charge qubit system. The proposed method uses a
continuous weak measurement model for charge measurements to estimate the bias control
pulses necessary to create a rotation to take the Bloch vector onto the x-axis of the Bloch
sphere. The method makes the purification process insensitive to rotations about the Bloch
sphere x-axis, arising from a constant Josephson tunnelling term.
1. Introduction
Superconducting charge qubits (Cooper pair boxes) are a promising technology for the realisation of
quantum computation on a large scale [1], where the logical ‘0’ and ‘1’ states are encoded using
localised charge states. For conventional fault-tolerant quantum computing, these quantum states
should have a high level of purity, preferably being as close to a pure state as possible. When the qubit
is coupled to an environment (simply treated as a source of noise) it is subject to decoherence which
will eventually turn the pure state into a completely mixed state – destroying the coherences that are
useful for quantum computing. However, if the environment represents a measurement device, the
evolution of the environment can be used to extract information about the evolution of the quantum
system and the resultant measurement record can be used to update the state of knowledge of the qubit
and increasing the purity of the qubit state – this is often referred to as a continuous ‘weak
measurement’ model [2]. The estimated qubit state can then be used to modify the behaviour of the
qubit through external controls, known as (Markovian) quantum feedback.
When using continuous weak measurements with low measurement strength, the time taken to
extract enough information about the underlying qubit state (evolving from a completely mixed state
to a satisfactory level of purity) can be considerable, given the small incremental nature of the weak
measurements. Interestingly, it is possible to use quantum feedback to increase the effective
purification rate [3, 4]. It has been discovered previously that the rate of increase in the average
purity is a maximum when the qubit Bloch vector is rotated onto the plane perpendicular to the
measurement axis, after each incremental measurement [3]. In this paper, this optimal protocol is
referred to as ‘ideal protocol I’, to distinguish it from an alternative protocol that aims to minimise the
average time taken to reach a given level of purity [5].
This paper addresses a complication which occurs when one attempts to apply this optimal method
to a realistic model of a superconducting charge qubit – the Cooper pair box [6]. The Cooper pair box
consists of a small island of superconducting material that is connected to the bulk material by a
Josephson junction, which allows the tunnelling of Cooper pairs on to and off the island. The problem
is that the sizes of the controls that can be applied in a real system are limited. The tunnelling gives
rise to a σx Hamiltonian corresponding to a rotation about the x-axis and a bias voltage can be applied
to generate a σz term. In the simplest Cooper pair box, the Josephson tunnelling frequency is fixed at
manufacture and controls can only be applied via the bias voltage/σz term. We will demonstrate that
significant improvements in the average purification rate should be achievable using this voltage bias
only.
2. System model
We have simulated the dynamics of a superconducting charge qubit, which consists of a small island
of superconducting material connected via a Josephson junction to a bulk superconducting electrode.
The electrode supplies a voltage bias, which can be as an effective charge ng. The island is also
capacitively coupled to a grounded electrode to supply a common reference. For simplicity, we ignore
the dynamics of the biasing circuitry [7].
V
CJ
CG
GND
V
CJ
CG
GND
CP
CP
Figure 1. Cooper pair box topology and equivalent electrical circuit network.
Table 1. Parameter values in line with experimental values taken from reference [8]
Symbol
EJ
CJ
CG
CP
γ
Description
Josephson junction energy
Josephson junction capacitance
Qubit-Grounded Bulk capacitance
Electrodes parasitic capacitance
Measurement strength constant
The Hamiltonian of this system is:
H
2e2  n 2  n

2Cq 
g
Value
10 GHz
500 aF
0.5 aF
1.0 aF
75×106
1  2e  1


 I 
x
  ng  z 
2
2Cq  2
2

2
g
(1)
The capacitance Cq is the effective qubit capacitance [7] calculated from the three physical
capacitances, CJ, CG and CP. Note: Cq ≈ CJ, where CJ is the capacitance of the Josephson junction, CG
is the electrode gate capacitance and CP is the parasitic capacitance formed between the two biasing
electrodes. The Hamiltonian is approximated by a two state system and a bias term ng is introduced to
represent the applied voltage as an effective charge as a number of Cooper pairs [6].
The first term of equation (1) may be discarded as the identity matrix does not affect the dynamics
of the system, but is included initially for completeness. The second term shows that the applied
voltage bias field controls the rotations about the z-axis (via σz). The speed, and direction of which is
governed by the magnitude of the bias; when ng = 0.5 the rotations are halted. The final term of
equation (1) is due to the Josephson tunnelling between adjacent charge states. In the Bloch sphere
representation this is equivalent to rotations around the x-axis, the frequency of which is fixed by
manufacture (in this case we take the frequency to be 10 GHz, in line with experimental values [8]).
3. Weak measurement
When implementing quantum algorithms, it is preferable to work with pure states because gate
operations such as rotations will have maximum effect. A pure qubit state is any state on the surface of
the Bloch sphere. Those states not on the Bloch sphere surface, but inside the sphere, are mixed states.
The mixed state is usually a result of decoherence of a pure state, the long time effect of this
decoherence is to attract the Bloch vector to the centre of the sphere, creating a completely mixed
state. Gate operations have no effect on this completely mixed state hence it is generally not suitable
for quantum computation. Mixed qubit states are written in the density matrix formulation, as a 2×2
matrix ρ, where the off diagonal elements represent of the coherence of any superposition states. We
define the impurity by the following, [2]:
(2)
L  1  P  1  Tr  2
so that any pure state has an impurity of zero, and the completely mixed state has an impurity of 0.5.
The evolution of the density operator ρ in the presence of an environment is governed by the master
equation. Where the environment couples to the qubit charge, the environmental operator is
proportional to σz giving a master equation of the form
 
dˆ  


i ˆ

H , ˆ dt  ˆ z , ˆ z , ˆ dt


(3)
In the presence of an environment that performs a weak measurement, where the measurement record
is maintained, the density operator ρc is conditioned on the measurement record. This introduces a
stochastic term:
dˆ c  


i ˆ

2
ˆ z ˆc  ˆcˆ z  2 ˆ z ˆc dW
H , ˆ c dt  ˆ z , ˆ z , ˆ c dt 



(4)
where γ is the measurement strength and dW is a Weiner increment [2], a zero mean Gaussian
distributed random variable, with variance equal to the time step dt. The first term applies the
Hamiltonian dynamics to the density matrix, the second is a term representing the deterministic decay
of the mixed state towards the z-axis, and the third term is noise generated by measurement process. In
this context, the conditional density operator ρc represents the current state of knowledge of the
system, which is conditioned upon the measurement record produced in a specific experiment. The
accumulation of information/measurement record will purify the system because it removes the
uncertainty inherent in the (mixed) state of the system. As the state evolves under the action of weak
measurement, it is gradually pulled towards the surface of the Bloch sphere as information is extracted
(in the form of measurements). The fact that the information extracted by the measurement process is
dependent upon the orientation of the Bloch vector with respect to the measurement axis allows the
Hamiltonian to be manipulated by external controls so as to maximise the rate at which information is
extracted and hence the rate that the purity of the system increases.
The minimum rate of average purification occurs in a situation with no Hamiltonian evolution and
the qubit Bloch vector is allowed to drift stochastically towards the z-axis poles. In the case of very
weak measurement strengths the qubit state may remain near the centre of the Bloch sphere for
relatively long periods of time as there is little preference to continue to project towards a single pole.
This equation represents the minimum rate of increase in the average purity, which is used as a
baseline comparison for the performance metric defined by equation (13) below. Unfortunately, the
integral has no obvious analytic solution and so it has had to be solved numerically [3].
L 0 (t ) 
e 4t
8πt



exp(  x 2 /( 2t ))
cosh( 8 x)
dx
(5)
4. Feedback protocols
In this section we briefly explain the ideal protocol for achieving the optimal purification rate [3] and
why it would be difficult to implement in the Cooper pair box. We then propose a scheme whereby
most of the benefit of the ideal protocol can be obtained with the constraints present in the real system.
4.1. The ideal protocol
For a qubit subjected to a weak measurement process, it has been shown that quantum feedback can be
applied to enhance the rate of purification. This has been analysed in another paper [3] using a generic
qubit model, with instantaneous and perfect feedback. This work has provided an algorithm which
maximises the rate of increase in average purity. Equivalently, the maximum decrease in the average
impurity is obtained when the Bloch vector lies on the plane perpendicular to the measurement axis.
In the case of a measurement along the z-axis, this corresponds to continually rotating the Bloch vector
onto the xy-plane. By instantly and perfectly rotating the qubit back onto this plane after each
incremental time step, the average purity is maximised. The speed-up in purification tends to a factor
of two as the impurity tends to zero, this factor/ratio is defined by equation (13) below. Starting from
equation (4) the average incremental change in impurity can be obtained:


dL  8 dtL 1  1  2 L  cos 2  
(6)
where θ is the angle between the Bloch vector and the measurement axis prior to a measurement step.
When θ = 90o the cosine term is removed and the size of the increment is maximised.
dL optimum  8 dtL
L optimumt   e8 t L0 
(7,8)
thus providing the fastest reduction in the average impurity when the Bloch vector is perpendicular to
the measurement axis. Equation (8) forms the upper bound for the performance metric given in
equation (13).
However, it would be extremely difficult to apply these instant and perfect control fields in a
practical qubit. In addition, for the model qubit considered here, there is also the rotation of the Bloch
vector around the x-axis, due to the non-zero Josephson junction energy. The feedback protocol
proposed in this paper addresses these key issues.
4.2. Practical protocol
The algorithm proposed in this paper attempts to use finite duration voltage bias pulses to rotate the
Bloch vector repeatedly on to the x-axis, taking a screw-like path (see Figure 2). The x-axis is of
particular interest as it is invariant under x-rotation, therefore if the vector can somehow be positioned
close to the x-axis, it should remain close to the xy-plane, even in the absence of further control
pulses. This is why the x-axis is an attractive target in the presence of continuous x-rotation (due to
Josephson tunnelling). However, the effect of the weak measurement is to pull the Bloch vector away
from the xy-plane and towards the poles. The Bloch vector will be gradually pulled away from the xaxis in a growing spiral path. To successfully return the Bloch vector to the x-axis, a simple control
scheme has been devised which utilises a finite duration rotation about the z-axis to return the vector
to the x-axis within half a cycle of the rotation period.
Although it is important to remember that the weak measurement process creates measurement
noise, for simplicity it is not shown in figure 2. This noise corrupts the ideal path of the Bloch vector,
so that each experimental run will give a slightly different path and the timing of the control pulses
will vary slightly in accordance with the measurement record actually received. In general, it is
favourable to engineer a high frequency Josephson junction EJ for faster x-rotations ωx, as this is the
limiting factor that defines both the z-axis rotational frequency ωz and pulse duration τ. In addition, a
larger Josephson tunnelling frequency also increases the minimum energy level separation for the
qubit [4, 5], which tends to reduce the effect of thermal fluctuations on the evolution of the system.
Ignoring the noise effects of weak measurement at present, it is possible to take a point in the xzplane (Figure 2) as a start point and use a π-rotation about a tilted axis (dotted arrow) to finish exactly
on the x-axis by rotating from the very top to the very bottom of the circular path about this tilted axis.
The angle α of the axis can be calculated using elementary geometry from the measurement of the zaxis and the distance from the centre of the Bloch sphere R, which can be obtained from the
conditional density matrix ρ (which represents the current state of knowledge of the system).
Z



1
2
  sin 1 


1  2 P  
z LIMIT
x  2EJ
(9)
(10)
ZLIMIT
 1

 1
2
 cos 

Pure
z  x 
α
Mixed

X
1
2
2  x2   z2
(11)
(12)
Figure 2. The idealised path taken by the Bloch vector (if the measurement noise could be
removed). Ordinarily the vector spirals and lengthens in the yz-plane (dashed line). When the
vector exceeds ZLIMIT the feedback is triggered and a π-pulse is applied to rotate the vector down to
the x-axis.
The feedback process triggers when the positive peak Z measurement result exceeds a particular
threshold, ZLIMIT (Figure 2). On triggering, the system calculates the z-axis rotational frequency ωz to
tilt the plane of rotation by angle α, such that the lowest point of the tilted circular path coincides with
the x-axis. The speed of the z-rotation is set by the bias value ng, and a square pulse maintains a
constant bias for the required pulse duration τ. A significant advantage of this approach is that the
control field does not need to be continually altered in the presence of the stochastic measurement
noise. After the Bloch vector has reached the vicinity of the x-axis, the z-rotation is removed, so the
qubit only experiences the constant x-rotation, this creates a spherically distorted spiral in the yz-plane
(simplified as blue dashed lines in figure 2). This gradually growing spiral will eventually exceed
ZLIMIT where again the feedback will trigger. The overall effect is to constrain the Bloch vector to the
region Z ≤ ZLIMIT near the xy-plane.
5. Results
Figure 3 shows the improvement or ‘speed up’ in the average purification rate [3], as defined by
equation (13), which is given as the ratio of the time taken for the average impurity to reach a given
value under the action of pure measurement (determined by the numerical solution of the integral in
equation 5) and under the protocol employed here. As the performance measure S is a ratio of two
purification times, this graph is independent of the measurement strength γ. It should also be noted
from the shape of the graph that the performance increase is not constant for all values of purity, with
maximum gains obtained at high purity (the final part of the time evolution). The optimal
improvement given by the ideal protocol tends to a factor of two as the limit of the remaining impurity
is taken to zero.
As can be seen in figure 3, the practical protocol proposed in this paper performs well (dashed
line), almost reaching the same performance of the optimal ideal protocol (solid line). The result
shown in figure 3 also includes errors to due inaccuracies in determining the times at which the Bloch
EMBED
Equation.3
L worst t  

e  4 t
8 t



 dx
exp  x 2 2t 

cosh 8 x

vector passes the ZLIMIT threshold, indicating that the performance of the practical purification protocol
is quite robust to common sources of error.
S
 

T L0
T L
(13)
Figure 3. The improvement in the average purification rate as a function of the remaining impurity
for ZLIMIT = 0.333.
6. Conclusions
We have proposed a practical implementation of a rapid state purification protocol [3]. The ideal
protocol maximises the rate of increase of the average purity of the system but requires perfect and
instantaneous control over the system Hamiltonian. The protocol considered here is more practical for
situations where the controls are limited and the natural (no feedback) Hamiltonian evolution is
significant. For the specific example considered in this paper, that of a Copper pair box subject to
weak charge measurements, the practical protocol uses simple π pulses of the available control field
(voltage bias) to rotate the Bloch vector on to the x-axis once it passes a fixed threshold value. This
yields a significant improvement in the rate of increase of the average purity, very close to the ideal
protocol.
Acknowledgements
EJG would like to acknowledge the support of the Department of Electrical Engineering and
Electronics, and a University of Liverpool research scholarship. CH and JFR would like to
acknowledge the support of an ESPRC grant: EP/C012674/1.
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