Download Guidance Applied to Quantum Operations in Josephson

Guidance Applied to Quantum Operations in Josephson
Charge Qubits
Charles D. Hill, Jason F. Ralph, Elias J. Griffith
Department of Electrical Engineering and Electronics, Brownlow Hill, Liverpool, L69
3GJ, United Kingdom.
[email protected]
Abstract. In this paper we present an adaptation of classical guidance laws to quantum
operations. One such guidance technique which has proved particularly useful in classical
systems is proportional navigation. Despite the fact that proportional navigation controls are
not optimal, controls based on these methods are often simpler, easier to implement and have a
lower bandwidth than those corresponding to optimal controls. We consider the use of
proportional navigation laws for quantum control. These techniques have previously been
considered for single qubit state preparation but not for qubit gate operations [1]. This paper
extends these techniques, showing how classical proportional navigation can be applied to the
problem of implementing quantum operations using examples drawn from solid-state
Josephson charge qubits.
1. Introduction
The coherent operation and control of quantum mechanical systems is typically controlled by the
application of classical external bias fields. These fields are subject to noise which will then couple to
the system, limiting the coherence of the quantum mechanical system. In a classical system, a closed
control loop is often used reduce the unwanted effects of noise. Closed loop control compares the
desired state of the system with the measured state of the system and minimises the error between the
two. In a quantum mechanical system, this is problematic because measurement inevitably causes a
back action on the system being measured. Nevertheless, several groups have proposed protocols
based on weak measurement which minimize the adverse effects of the back action on the system,
while still providing a measurement record suitable for feedback [2,3].
In this paper we generalize a classical guidance technique to a quantum mechanical system. We
show how it is possible to use guidance techniques to find the classical bias fields for the
implementation of a given quantum operation for a superconducting charge qubit. Although the
method we consider here is an example of open loop control, one of its principle advantages is that the
method easily generalises to feedback. This is because the guidance technique which we use to find
the controls is much simpler, and faster, than corresponding optimal control methods. For many
applications it would be impractical to recalculate an optimal pulse shape in response to feedback.
With the simpler guidance methods adapting to feedback becomes more feasible. So, although
guidance techniques are not time optimal, they offer significant advantages.
This paper is organised as follows. Section 2 describes proportional navigation, and gives a method
for finding the quantum mechanical analogue. Following this prescription it is comparatively simple to
find the bias controls which implement a given unitary gate in a specified time. In Section 3 we
consider the application of quantum guidance, and demonstrate how these techniques can be applied to
a model superconducting charge qubit. Finally, conclusions are made in Section 4.
2. Proportional Navigation Guidance
Proportional navigation is a control technique that ensures that a specified target state is reached at a
specified time. As the system evolves there is always an error between the desired evolution and the
actual evolution. Controls are applied to reduce the distance between desired evolution and the actual
evolution. In proportional navigation the controls applied are proportional to the Zero Effort Miss
(ZEM), a measure of the ‘distance’ the system would miss the target state by if no controls were
applied. The most general form of classical proportional navigation is:
ac  N '
( ZEM )
2
t go
where ac is the acceleration required or ‘commanded’, N’ is a constant known as the navigation
constant and typically chosen between 4 and 8. ZEM is the Zero Effort Miss. The controls are
moderated by 1/tgo2 where tgo is the time to go until the end of the operation. Classically proportional
navigation is not time optimal, but does have some desirable features which has lead to its widespread
use in classical systems.
Several of the important features of proportional navigation (in any context, classical or quantum)
can be seen in the Figure 1. At time t, we find ourselves in the state U(t). If no controls are applied the
system will evolve from the current state, U(t), to the state UEnd. The ZEM is a measure of distance
between UEnd and the intended objective, UObjective. Controls are applied to the system to minimize the
ZEM. In proportional navigation, the controls are proportional to the Zero Effort Miss. Up until this
point the states, U, could have referred to any state, quantum or classical. From now on we will
consider a particular case: that each state U refers to a particular unitary operation.
Figure 1:
Schematic diagram showing the
essential features of proportional
navigation.
We now show how it is possible to adapt proportional navigation to the problem of implementing a
given quantum operation, UObjective. In adapting the classical guidance algorithm to the quantum
mechanical setting, we have to keep in mind that operations do not commute. Applying no controls,
we reach the operation,
U End  U 0 (t go ) U (t )
where U0 is the natural evolution of the system (with no controls applied). That is
 iH t 
U 0  exp   0  , where H0 is the Hamiltonian of the system when no controls are applied. The
 

†
operation which characterises the ZEM is given by the unitary operator, U ZEM  U ObjectiveU End
.
Because operations do not commute, we have to retrodict operations from the end of the evolution, to
find the correct controls to apply now, rather than at the end of the operation. That is,
U now  U 0† U ZEM U 0
If it were possible to instantly apply Unow instantaneously, then no more controls would need to be
applied to reach the objective because,
U Objective  U 0 (t go ) U now
However, since it is not possible in many practical situations to apply arbitrarily strong and
instantaneous controls we must cancel the error signal, given by Unow, over a period of time. We now
adapt proportional navigation to the quantum mechanical regime to specify a particularly simple and
effective way of canceling the error, Unow. This gradual cancellation of errors is one of the most
appealing aspects of proportional guidance, because controls are applied gradually over a period of the
bandwidth of the controls tends to be low and the controls tend to be small immediately before
reaching the objective.
Proportional navigation can be adapted to quantum mechanics by using three steps. These are:
1. Parameterise the operation according to angles of rotation. For one qubit this is,
 i  Tr  i log( U now )
i
2
2. Attempt to apply the control Hamiltonian given by the quantum analogue of classical
proportional navigation:
H i  N '
i
t go
3. Repeat for each time-step.
Using these three steps it is often possible to find controls which reach the given target state, UObjective,
for a large number of practical situations. We did not assume typically applied controls before we
started, such as resonant fields, but find that these controls are found as a result of the simple guidance
law given in step 2. This quantum guidance law differs as it is proportional to 1/tgo, and not 1/tgo2 as it
is in the classical case. That change is because controls applied to the Hamiltonian affect first order (in
time) angular precession frequencies directly, rather than accelerations as the controls do in the
classical case.
3. Application to a model charge qubit
We now demonstrate quantum guidance as it could be applied to an idealised model for a standard
experimental configuration of the superconducting charge qubit. This qubit consists of a single
superconducting island coupled to external circuitry via two Josephson junctions (see Figure 2). The
control fields which can be applied to this system are a voltage bias, Vx, and an external flux, Φx. The
two Josephson junctions form a loop. The flux through the loop, Φx, modulates the rate of electron
pairs tunnelling onto and off the superconducting island. For simplicity, in our idealised system, we
have assumed that both Josephson junctions are identical. With this model in place, we wish to see
how proportional navigation can be used to find a given unitary operation of the charge qubit state,
where the basis states are given by no excess electron pairs (|0) or one excess pair (|1) on the Cooper
pair box.
Figure 2:
Circuit diagram of a
single Josephson
charge qubit.
Figure 3:
Energy level diagram,
showing the variation in
energy levels at different
voltage biases, and
Φx=(0.25±0.05)Φ0.
Typical values in this idealized qubit were taken to match experiment, in particular those values
found in References [4, 5]. Figure 2 shows a schematic diagram of a qubit circuit, with total
capacitance C = C1+C2, a bias voltage of Vx and a flux of Φx. Typical values are used for this paper
are: C = 6×1016 F, ν/2π=12.9GHz, Φx=0.25Φ0 ±0.05Φ0, where Φ0=2×1015Wb, and Vx=(0.1 ± 0.04) ×
2e/C. For the purposes of this paper, we limit the variation available in each control as indicated. We
do this for practical reasons. For example, if the voltage bias becomes too large, then we will begin to
excite higher excited states, leading to leakage errors and so the voltage bias must be limited to
prevent this. Another criterion that we would like to satisfy is to keep the strength of the control linear
in the control field, which limits the variations which we can apply to the flux. When these limits are
applied we can limit ourselves to considering just two available states.
When limited to just two states, the Hamiltonian of the charge qubit may be written as
 CVx2 e 2 
 π x 
h
 I  eVx z 
 x
H 0  

cos

2
2
C
2

 0 


where σx and σz are the Pauli sigma matrices, and typical values for each of the experimental
parameters is given above. The corresponding energy level diagram is shown in Figure 3. The first
term in the Hamiltonian corresponds to a global phase, and can safely be ignored in the one qubit case.
The second term represents an electrostatic energy splitting of the energy levels proportional to the
applied external voltage. The third term corresponds to tunneling on and off the island, and can be
modulated by application of flux through the loop formed by the two Josephson junctions. This gives
us two external controls which affect the evolution of the charge qubit. By applying time dependant
pulses to these two controls (or indeed either one) it is possible to perform every possible rotation of a
single qubit. In this paper we will consider an operation which cannot be applied directly, but for
which we require time dependant controls – a rotation by π around the y-axis.
In order to implement proportional navigation we would like to find the desired controls we make
updating the controls relatively simple. To do this, we use a linear approximation:
2H x
 
π sin  π x
 0
H
V x  z
e
 x 



If we keep within the limits which we have specified for the flux and voltage controls, then this
approximation is valid. Using the linear approximation, and steps 1 and 2 of the proportional
navigation prescription, we can find the controls which should be applied to the qubit required to
implement a given single qubit rotation. The controls which are obtained for a rotation by π around the
y-axis are shown in Figure 4.
(a)
(b)
Figure 4: These diagrams show (a) the controls which should be applied using proportional navigation,
and (b) the corresponding Zero Effort Miss.
Figure 4 shows the controls and zero effort miss calculated using proportional navigation for the
implementation of a rotation by π around the y-axis for a superconducting charge qubit. These pulses
are typical of the pulses found when using proportional navigation to implement a quantum operation.
Even though it is not possible to rotate around the y-axis directly, it is possible to create a Y operation
by using time varying flux and voltage biases. Figure 4(a), the controls shows that the controls can
saturate, when the correction which needs to be made is large, but as the evolution continues, the angle
which needs to be corrected dies away (as is shown in Figure 4(b)) and therefore so do the
corresponding controls. This is a feature typical of proportional navigation. Controls die away towards
the end of the evolution. This means that as we finish applying a given operation, we do not need to
make abrupt changes in the strength of the control fields, and potentially populating higher energy
excited states. One advantage of proportional navigation is that it leads to a stationary Hamiltonian
towards the end of the evolution.
In order to compare the two methods, the corresponding optimal control pulse, found using
gradient ascent methods [6, 7], enforcing similar limits on the controls which we imposed on the
proportional navigation controls. The optimal control takes 200ps while the proportional navigation
controls nominally take 750ps. Without these constraints optimal pulses can be as short as 40ps.
However the proportional navigation controls can be easily adapted, whereas the optimal controls can
take some time to calculate.
Proportional navigation techniques can be made comparatively robust against low bandwidth
constraints, because they can be easily adapted to feedback. In Figure 5, a first order time delay has
been added to the system:
f (t )  f (t ) * exp( t /  )
The dotted and dot-dashed lines show the trace fidelity of the operation for proportional navigation
and optimal methods when no correction is applied. The optimal pulse scheme marginally outperforms
the proportional navigation algorithm. In contrast both are outperformed by the solid line which shows
the fidelity of the proportional navigation pulse when the pulse uses feedback to compensate for the
imperfect pulses applied to the system if we had perfect knowledge of the actually applied pulses.
Using this method, proportional navigation could, in principle, compensate for time constants as large
as 8ps. If the optimal pulse or guidance pulse is naively used the fidelity drops off almost immediately.
Figure 5: The effect of adding a first order time delay to the
optimal, and proportional navigation techniques.
There are several advantages to using proportional navigation. It is a much simpler method than
finding optimal pulses. Controls die away towards the end of the evolution, leading to a stationary
Hamiltonian at the end of each pulse sequence, and it easily generalises to feedback. Some
disadvantages of the method are that it is not optimal – faster controls are possible at the expense of
complexity, and it requires shaped pulses.
4. Conclusion
In this paper we have adapted classical guidance techniques to apply to quantum operations. A
prescription was given for the quantum analogue to proportional navigation, and we showed how
quantum guidance could be applied to an idealized superconducting charge qubit to implement a given
quantum operation (in this paper we considered the Y operation). Whereas these pulses are not as fast
as the corresponding optimal pulse shapes (which we explicitly found), they are much simpler, easier
to find and adapt and in many cases they may prove sufficient. These pulses can adapt to low
bandwidth situations, and easily generalize to a feedback situation. In practise, these control methods
have proven useful in the classical regime, and there is reason to believe that they would be equally
useful in the quantum regime.
Acknowledgments
CH and JFR would like to acknowledge the support of an ESPRC grant: EP/C012674/1. EJG would
like to acknowledge the support of the Department of Electrical Engineering and Electronics, and a
University of Liverpool research scholarship.
References
[1] Ralph, Griffith, Clark, Everitt, ‘Guidance in a Josephson Charge Qubit’, Physical Review B,
70, 214521 (2004).
[2] V.P.Belavkin, Rep. Math. Phys. 45, 353 (1999).
[3] A.C.Doherty, K.Jacobs, ‘Feedback control of quantum systems using continuous stateestimation’, Proc. 39th IEEE Conf. on Decision and Control, p.949 (2000)
[4] Pashkin, Yamamoto, Astafiev, Nakamura, Averin, & Tsai, ‘Quantum oscillations in two
coupled charge qubits', Nature 421 (6925), 823—826, (2003).
[5] Yamamoto, Pashkin, Astafiev, Nakamura, & Tsai, 'Demonstration of conditional gate operation
using superconducting charge qubits', Nature 425 (6961), 941—944, (2003).
[6] Khaneja, Reiss, Kehlet, Schulte-Herbruggen, Glaser, ‘Optimal control of coupled spin
dynamics: design of NMR pulse sequences by gradient ascent algorithms’, Journal of Magnetic
Resonance, 172, 296-305, (2005)
[7] Spoerl, Schulte-Herbrueggen, Glaser, Bergholm, Storcz, Ferber, Wilhelm, ‘Optimal Control of
Coupled Josephson Qubits’, quant-ph/0504202, (2005).