Rapid state purification in a superconducting charge qubit
11123E J Griffith, C D Hill, J F Ralph, K Jacobs and H M Wiseman
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.
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.
Superconducting charge qubits (Cooper pair boxes) are a promising technology for the realisation of quantum computation on a large scale , 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 . 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 . 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 .
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 . 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 σ Hamiltonian corresponding to a rotation about the x-axis and a bias voltage can be applied x
to generate a σ term. In the simplest Cooper pair box, the Josephson tunnelling frequency is fixed at z
manufacture and controls can only be applied via the bias voltage/σterm. We will demonstrate that z
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 n. The island is also g
capacitively coupled to a grounded electrode to supply a common reference. For simplicity, we ignore the dynamics of the biasing circuitry .
C C CCJGJGV GND V GND
Figure 1. Cooper pair box topology and equivalent electrical circuit network.
Table 1. Parameter values in line with experimental values taken from reference 
Symbol Description Value
E Josephson junction energy 10 GHz J
C Josephson junction capacitance 500 aF J
C Qubit-Grounded Bulk capacitance 0.5 aF G
C Electrodes parasitic capacitance 1.0 aF P 6γ Measurement strength constant 75×10
The Hamiltonian of this system is:
222121?ee;；;；;?，?，2 (1) HnnIn~~，？？？？？，?，?gggzx22222CC????qq
The capacitance C is the effective qubit capacitance  calculated from the three physical q
capacitances, C, C and C. Note: C ? C, where C is the capacitance of the Josephson junction, C JGPqJJG
is the electrode gate capacitance and C is the parasitic capacitance formed between the two biasing P
electrodes. The Hamiltonian is approximated by a two state system and a bias term n is introduced to g
represent the applied voltage as an effective charge as a number of Cooper pairs . 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 σ). The speed, and direction of which is z
governed by the magnitude of the bias; when n = 0.5 the rotations are halted. The final term of g
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 ). 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?(；
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 σ giving a master equation of the form z
；iˆˆˆˆˆˆ！，！，！，d；，？H,；dt？~,~,；dt (3) zz;;
In the presence of an environment that performs a weak measurement, where the measurement record is maintained, the density operator ρ is conditioned on the measurement record. This introduces a c