An adiabatic method for a single-shot non-demolition measurement of the phase qubit is suggested. The qubit is inductively coupled to a low-frequency resonator, which in turn is connected with a classical measurement device (phase meter). The resonator drives adiabatic oscillations of the supercurrent in the qubit loop. The back reaction of the qubit loop on the resonator depends on the qubit state. Measuring the phase shift of the resonator’s oscillations one can determine the state of the qubit. Numerical computations with available experimental parameters show that the phase difference between the two qubit states increases at a rate of 0.0044 rad/ns with the fidelity of about 0.9989 and the measurement time of about 100 ns. The fidelity of the measurement is estimated taking into consideration possible quantum transitions inside and outside the qubit manifold. An increase of the phase difference is possible but it is linked to a reduction of the fidelity. The requirements for the reproducibility of the qubit and resonator parameters are formulated.
An adiabatic method for a single-shot non-demolition measurement of the phase qubit is suggested. The qubit is inductively coupled to a low-frequency resonator, which in turn is connected with a classical measurement device (phase meter). The resonator drives adiabatic oscillations of the supercurrent in the qubit loop. The back reaction of the qubit loop on the resonator depends on the qubit state. Measuring the phase shift of the resonator’s oscillations one can determine the state of the qubit. Numerical computations with available experimental parameters show that the phase difference between the two qubit states increases at a rate of 0.0044 rad/ns with the fidelity of about 0.9989 and the measurement time of about 100 ns. The fidelity of the measurement is estimated taking into consideration possible quantum transitions inside and outside the qubit manifold. An increase of the phase difference is possible but it is linked to a reduction of the fidelity. The requirements for the reproducibility of the qubit and resonator parameters are formulated.
We analyze the dynamics of a qubit-resonator system coupled with a thermal bath and external electromagnetic fields. Using the evolution equations for the set of Heisenberg operators that describe the whole system, we derive an expression for the resonator field, that includes the resonator-drive, the resonator-bath, and resonator-qubit interactions. The renormalization of the resonator frequency, caused by the qubit-resonator interaction, is accounted for. Using the solutions for the resonator field, we derive the equation that describes the qubit dynamics. The dependence of the qubit evolution during the measurement time on the fidelity of a single-shot measurement is studied. The relation between the fidelity and measurement time is shown explicitly. We proposed a novel adiabatic method for the phase qubit measurement. The method utilizes a low-frequency, quasi-classical resonator inductively coupled to the qubit. The resonator modulates the qubit energy, and the back reaction of the qubit causes a shift in the phase of the resonator. The resonator phase shift can be used to determine the qubit state. We have simulated this measurement taking into the account the energy levels outside the phase qubit manifold. We have shown that, for qubit frequencies in the range of 8-12GHZ, a resonator frequency of 500 MHz and a measurement time of 100 ns, the phase difference between the two qubit states is greater than 0.2 rad. This phase difference exceeds the measurement uncertainty, and can be detected using a classical phase-meter. A fidelity of 0.9999 can be achieved for a relaxation time of 0.5 ms. We also model and simulate a microstrip-SQUID amplifier of frequency about 500 MHz, which could be used to amplify the resonator oscillations in the phase qubit adiabatic measurement. The voltage gain and the amplifier noise temperature are calculated. We simulate the preparation of a generalized Bell state and compute the relaxation times required for achieving high fidelities. We consider two capacitively coupled phase qubits similar to that used in the experimental work of R. Bialczak et al. (Nature Physics, 6, 409, 2010). The Bell state was created using a NOT gate and a universal entangling gate. We solved the equation of motion for the density matrix with the Bloch relaxation times, T{sub 1} and T{sub 2}. For every qubit we took into consideration all the states in the shallow well including the states outside the qubit manifold. Also we took into consideration unwanted interaction between the qubits during the application of the NOT gate. The NOT gate was implemented using a Gaussian pulse assuming the frequency difference between the two qubits is 200 MHz. An entangling gate was implemented reducing this frequency difference to zero. We have found the optimal parameters for the Gaussian pulse and the optimal duration for the entangling gate. We have shown that the maximum fidelity of 0.99 can be achieved for the minimum relaxation time, T{sub 1} = 1.2 {micro}s, if T{sub 2} = 2T{sub 1}. By slightly increasing the required value of T{sub 1}, one can sharply reduce the required value for T{sub 2}.
The realization of quantum adiabatic dynamics is at the core of implementations of adiabatic quantum computers. One major issue is to efficiently compromise between the long time scales required by the adiabatic protocol and the detrimental effects of the environment, which set an upper bound to the time scale of the operation. In this work we propose a protocol which achieves fast adiabatic dynamics by coupling the system to an external environment by the means of a quantum-non-demolition (QND) Hamiltonian. We analyse the infidelity of adiabatic transfer for a Landau-Zener problem in the presence of QND measurement, where the qubit couples to a meter which in turn quickly dissipates. We analyse the protocol's fidelity as a function of the strength of the QND coupling and of the relaxation time of the meter. In the limit where the decay rate of the ancilla is the largest frequency scale of the dynamics, the QND coupling induces an effective dephasing in the adiabatic basis. Optimal conditions for adiabaticity are found when the coupling with the meter induces dissipative dynamics which suppresses unwanted diabatic transitions.
Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi oscillations in a phase qubit. The external source, typically in the GHz range, induces transitions between the qubit levels. The resulting Rabi oscillations of supercurrent in the qubit loop are detected by a high-quality resonant tank circuit, inductively coupled to the phase qubit. Here we present the results of detailed computer simulations of the interaction of a classical object (resonant tank circuit) with a quantum object (phase qubit), which explicitly account for the back action of a tank circuit. According to the results of our simulations the Rabi oscillations in the MHz range can be detected using the conventional NMR pulse Fourier technique.
Fast, precise, and quantum-non-demolition (QND) readout of superconducting qubits is a fundamental component of high-fidelity quantum sensing and computation. Conventional approaches typically operate in the dispersive regime, where the qubit-resonator coupling g is weak compared to the detuning Δ. While exhibiting good QND properties, the readout rate is limited to ∼g2N/Δ≪g, where N is the number of photons in the resonator. QND readout in the nondispersive regime, where the readout rate reaches its full potential ∼g, relies on parameter sweeps that may encounter resonances, leading to measurement-induced state transitions (MIST). In this work, we study a nondispersive readout protocol that replaces these sweeps by sudden quenches of the coupling constant, using a resonator that is preloaded with photons. We call this protocol bang-bang readout, and show that it realizes single-shot projective measurements. The fidelity and QNDness of the qubit post-measurement are remarkably high, with an error that decreases like 1/N. We show that the protocol can also be implemented without preloading the resonator by instead strongly driving the qubit, e.g., with a classical flux drive.
Time-domain observations of coherent oscillations between quantum states in mesoscopic superconducting systems have so far been restricted to restoring the time-dependent probability distribution from the readout statistics. We propose a method for direct observation of Rabi oscillations in a phase qubit. The external source, typically in GHz range, induces transitions between the qubit levels. The resulting Rabi oscillations of supercurrent in the qubit loop induce the voltage oscillations across the coil of a high quality resonant tank circuit, inductively coupled to the phase qubit. It is the presence of these voltage oscillations in the detected signal which reveals the existence of Rabi oscillations in the qubit. A detailed calculation for zero and nonzero temperatures are made for the case of persistent current qubit. According to the estimates for decoherence and relaxation times, the effect can be detected using conventional rf circuitry, with Rabi frequency in the MHz range.
We have studied the possibility of a single-shot non-demolition measurement of a superconducting qubit using a microstrip SQUID amplifier (MSA). The Johnson noise generated by all resistors in the MSA is taken into consideration. We show that a single-shot non-demolition measurement is possible with six photons in the measurement resonator. For a phase qubit inductively coupled to a measurement resonator we have obtained the expression for the mutual inductance required for measurement of the qubit state.