Incoherent qubit control using the quantum Zeno effect

TL;DR

This paper demonstrates a novel method of qubit control using the quantum Zeno effect through continuous measurement, enabling error detection and high-fidelity state tracking without direct Hamiltonian manipulation.

Contribution

It introduces a measurement-only control technique employing a dynamically controllable observable and continuous monitoring, advancing quantum feedback and state purification methods.

Findings

Successfully demonstrated state control via the quantum Zeno effect in circuit-QED.

Achieved high-fidelity state tracking through post-selected measurement records.

Provided a new tool for quantum feedback protocols and rapid state purification.

Abstract

The quantum Zeno effect is the suppression of Hamiltonian evolution by repeated observation, resulting in the pinning of the state to an eigenstate of the measurement observable. Using measurement only, control of the state can be achieved if the observable is slowly varied such that the state tracks the now time-dependent eigenstate. We demonstrate this using a circuit-QED readout technique that couples to a dynamically controllable observable of a qubit. Continuous monitoring of the measurement record allows us to detect an escape from the eigenstate, thus serving as a built-in form of error detection. We show this by post-selecting on realizations with arbitrarily high fidelity with respect to the target state. Our dynamical measurement operator technique offers a new tool for numerous forms of quantum feedback protocols, including adaptive measurements and rapid state purification.

Equations

$$H=\frac{\chi {\bar{a}}_0}{2}(a+a^{†}){\sigma }_{\delta (t)},$$

$$d\rho =\frac{{\Gamma }_D}{2}\mathcal{L}[{\sigma }_{\delta (t)}]\rho dt+\sqrt{\frac{{\Gamma }_D}{2}\eta }\mathcal{H}[{\sigma }_{\delta (t)}]\rho dW,$$

$$d\rho$$

$$\theta =\arctan \left(\frac{2\Omega }{{\Gamma }_D+\sqrt{{\Gamma }_D^2-4{\Omega }^2}}\right).$$

$$\Omega (V)$$

$$\rho (t+dt)$$

$$\eta =\frac{({\mu }_{y=+1}-{\mu }_{y=-1}{)}^2}{8\tau {\sigma }^2{\Gamma }_D},$$

Full text

In the field of quantum control, two essentially distinct resources are available for state manipulation. Application of a time-dependent Hamiltonian via external driving enables state preparation given a known initial state. In contrast, measurement and dissipation provide a uniquely quantum resource, owing to the stochastic back-action that necessarily accompanies acquisition of information. In addition, measurement-based, or incoherent control Wiseman and Milburn (2009) also extracts entropy from a system, this information can be used to detect and correct for errors and imperfections. While incoherent and Hamiltonian control are often used in conjunction Motzoi et al. (2015); Martin et al. (2015); Thomsen et al. (2002); Jacobs (2003); Riste et al. (2013); Katz et al. (2008), full control is also possible using measurement alone Wiseman (2011); Ashhab and Nori (2010); Jacobs (2010); Roa et al. (2006); Aharonov and Vardi (1980); Pechen et al. (2006); Shuang et al. (2008). Measurement-only manipulation has been demonstrated using a fixed measurement basis Blok et al. (2014), but unlike Hamiltonian-based methods, implementation of a time-dependent measurement basis is lacking. Such a capability is a versatile additional degree of freedom for measurement based protocols, such as rapid state purification Jacobs (2003) and state manipulation Tanaka and Yamamoto (2012); Ashhab and Nori (2010); Jacobs (2010), for control by projection into a subspace referred to as quantum Zeno dynamics Burgarth et al. (2014a), and for measurement-based quantum computation Raussendorf and Briegel (2001).

In this Letter, we present a method to dynamically tune the measurement operator in a circuit-QED system, and use this capability to deterministically and incoherently manipulate the state of an effective qubit. Our method relies on the suppression of coherent evolution via strong measurement, known as the quantum Zeno effect (QZE), which has been observed in many systems Kwiat et al. (1999); Hosten et al. (2006); Bernu et al. (2008); Itano et al. (1990); Gagen and Milburn (1993); Nagels et al. (1997); Fischer et al. (2001); Streed et al. (2006); Wolters et al. (2013); Palacios-Laloy et al. (2010); Kakuyanagi et al. (2015); Slichter et al. (2016). Essentially it pins a quantum state to an eigenstate of the measurement operator. Changing the operator at a rate slow compared to the rate of measurement-induced dephasing $\Gamma_{D}$, we effectively ‘drag’ the state using measurement alone Aharonov and Vardi (1980); Pechen et al. (2006); Shuang et al. (2008); Roa et al. (2006). This method does not require the measurement record or feedback to achieve control. However by monitoring the record with a quantum-limited Josephson parametric amplifier (JPA), we characterize the dynamics and verify good agreement with theory. In the fast-driving limit, where the Zeno effect breaks down, we observe a characteristic arcing effect in which the state maintains relatively high purity even as is transitions to the unwanted measurement eigenstate. Using the measurement record to post-select, we show that we can achieve arbitrarily high fidelity with respect to the target state. Thus measurement serves a dual role, both controlling the state and providing real-time information on its performance.

Our system setup is similar to the one used in Ref. Hacohen-Gourgy et al. (2016). It consists of a transmon Koch et al. (2007); Paik et al. (2011) qubit dispersively coupled to the modes of a 3D superconducting cavity. We apply a tone resonant with the qubit frequency that drives Rabi oscillations on the qubit at a frequency of $\Omega_{R}$, so that its Hamiltonian becomes that of an effective qubit with energy splitting determined by $\Omega_{R}/2\pi$=40 MHz. The new energy eigenstates in this dressed basis are $\ket{\pm}=(\ket{g}\pm\ket{e})/\sqrt{2}$, where $\ket{g}$ and $\ket{e}$ are the ground and excited states of the bare qubit respectively. It is within the frame of this effective qubit that we demonstrate the ability to drag the state. We then apply a pair of sideband tones detuned above and below the cavity frequency by $\Omega_{R}$, as illustrated in Fig. 1, which gives us the following Hamiltonian for our effective low frequency qubit Hacohen-Gourgy et al. (2016),

[TABLE]

where $\bar{a}_{0}$ is the amplitude of sideband tones, $a$, $a^{\dagger}$ are the cavity ladder operators, and $\chi$ is the qubit dispersive frequency shift. The measurement operator $\sigma_{\delta(t)}\equiv\sigma_{x}\cos\delta(t)+\sigma_{y}\sin\delta(t)$ is set by the relative sideband phase $\delta(t)$. This Hamiltonian is a resonant cavity drive, the sign of which depends on the qubit state along the $\sigma_{\delta}$ axis. Detecting the cavity output field yields a measurement of the qubit at a rate $\Gamma_{M}=\Gamma_{D}\eta=2\chi^{2}\bar{a}_{0}^{2}\eta/\kappa$ in the $\sigma_{\delta}$ basis Gambetta et al. (2008), where $\kappa$ is the cavity mode decay rate and $\eta$=0.49 is the detection quantum efficiency. We detect the cavity displacement using a JPA operated in phase-sensitive mode, choosing the amplified axis to align with the displaced quadrature. The full system calibration procedure can be found in Ref. Hacohen-Gourgy et al. (2016).

We start by initializing the effective qubit in the $\ket{y=+1}$ state, which corresponds to the ground state of the non-driven transmon qubit. We then continuously measure the effective qubit while changing the measurement axis. This is followed by one of seven pulses $\{\text{I},x_{\pi/2},-x_{\pi/2},y_{\pi/2},-y_{\pi/2},x_{\pi},-x_{\pi}\}$, and a strong projective measurement for tomography. The dephasing rate during the continuous measurement is fixed, and set to $\Gamma_{D}/2\pi$ = 0.13 MHz. We repeat the runs for measurement rotation speeds relative to the effective qubit spanning from $v$=0.01 MHz to $v$=0.18 MHz, and perform tomography at intervals from 1$\mu$s to 5$\mu$s for each rotation speed. The thermal population of the transmon qubit was about 15%, so before each measurement we perform a 1$\mu$s projective measurement heralding the preparation state. We also use the projective readout at the end to ensure that the transmon qubit is still within the two-level subspace after the run. The tomography for the ensemble average behavior is shown in Fig. 2a. The colored dots show the tomography from $\sim$20000 traces per dot, and the lines are theory for the following parameters: initial state with $\langle y\rangle$=0.94, $\langle x\rangle=\langle z\rangle$=0, $\Gamma_{D}/2\pi$ = 0.13 MHz, and an additional pure dephasing, which we attribute mainly to instabilities in the Rabi drive, at a rate $\Gamma_{\phi}/2\pi$=0.005 MHz (corresponding to the decay time of the Rabi oscillations of the bare qubit). The statistical errors are negligible and the small discrepancy of the tomography data with theory is most likely due to systematic drifts of the measurement rate (amplitude of the side band tones) and leakage tone at the cavity mode frequency (LO leakage - see methods in Ref Hacohen-Gourgy et al. (2016)).

We now focus on the conditional dynamics of the state as it is being dragged. For this, we reconstruct the quantum trajectories Murch et al. (2013); Hacohen-Gourgy et al. (2016) from the continuous traces (see also the supplemental material). Our system is in a regime where $\kappa\gg\Gamma_{D}$, in which we can infer the diffusive nature of the quantum jumps. Because we operate the JPA such that it amplifies the optimal (informational) quadrature, the qubit evolution due to the measurement is not affected by phase back-action Gambetta et al. (2008); Korotkov (2001). Then the dynamics of the system can be described by the following master equation, in Itô form Wiseman and Milburn (2009); Jacobs and Steck (2006):

[TABLE]

where $\mathcal{L}[X]\rho=X\rho X^{\dagger}-(X^{\dagger}X\rho+\rho X^{\dagger}X)/2$ is the Lindblad dissipation superoperator, $\mathcal{H}[X]\rho=X\rho+\rho X^{\dagger}-\langle X\rho+\rho X^{\dagger}\rangle\rho$, and $dW$ is a Gaussian distributed variable with a variance $dt$ Øksendal (2003), which is itself extracted from the measurement record. We use the POVM that generates this equation with additional corrections to account for extra dephasing on the effective qubit (at a rate $\Gamma_{\phi}$) to reconstruct the trajectories as function of time from the continuous traces (see supplemental material). Fig. 2b shows two example trajectories for a dragging velocity of $v$=50KHz, with one trajectory showing a state that was successfully dragged, while the other illustrates a ‘quantum jump’. Note that after the jump the measurement process continues to drag the state on the opposite side of the Bloch sphere.

The dynamics of the whole ensemble can be visualized by plotting the distribution of the state of the qubit in the Bloch sphere as function of time, as shown in Fig. 3. There are several prominent qualitative features in these plots. As expected, the rate at which the qubit jumps is larger for faster dragging velocities. Strikingly, these quantum jumps always diffuse in an arc that extends opposite to the direction of rotation. This can be understood from the form of the back-action, which is zero at the poles of the measurement axis, and maximal in-between. Hence, when the state gets ‘pushed forward’ (that is, in the direction of the rotation) by the back-action, it is pushed towards a region of lower back-action. At the same time, it cannot go past the measurement axis because the back-action goes to zero at the pole. On the other hand, if the state gets ‘pulled back’ by the back-action, it is towards a region of higher back-action, thus having an increased probability of ‘escaping’ and undergoing a transition to the other side of the Bloch sphere, i.e. a quantum jump. Due to the relatively high quantum efficiency of our system, the state remains close to the surface of the Bloch sphere, and trajectories that jump arc out before arriving at the other side.

A consequence of the arcing feature in the dynamics is the lagging of the average of the state behind the measurement axis. For our specific experiment the ensemble averaged dynamics can be solved analytically by going into a frame rotating at the dragging velocity $v$, where the measurement axis is fixed and the qubit is driven by the Hamiltonian $H=(\Omega/2)\sigma_{z}$, with $\Omega=2\pi v$. In this measurement-axis frame the average qubit state evolves according to

[TABLE]

where the measurement axis is now fixed along the $y$ direction, and for simplicity we drop the negligible purely dephasing term $\Gamma_{\phi}$. The solutions display two characteristically different regimes:

• (i)

$\Gamma_{D}<2|\Omega|$ – oscillatory with $\lambda_{\pm}$ complex, and

• (ii)

$\Gamma_{D}\geq 2|\Omega|$ – overdamped with $\lambda_{\pm}$ real.

$\lambda_{\pm}=(-\Gamma_{D}\pm\sqrt{\Gamma_{D}^{2}-4\Omega^{2}})/2$ and $\vec{V}_{\pm}=\left(1,(\lambda_{\pm}+\Gamma_{D})/\Omega\right)$ are the eigenvalues and eigenvectors respectively. In the oscillatory regime the state of the qubit oscillates with respect to the measurement axis, and thus is not dragged by the measurement. In the overdamped regime, or Zeno regime, the oscillatory behaviour vanishes and is replaced by exponential decay along the axes defined by the eigenvectors $V_{\pm}$. As $\Gamma_{D}\rightarrow\infty$, the eigenvalue $\lambda_{+}$ goes to zero, which means that if the qubit starts near a pole of $\vec{V}_{+}$ it will remain pinned to it for an arbitrarily long time.

The slow decay for $\Gamma_{D}<\infty$ can be attributed to quantum jumps between the poles of $\vec{V}_{+}$. In each realization of the experiment these jumps can be observed. The jump axis is identified with the direction in which the damping rate is smallest, since the fast damping in the orthogonal direction aligns the poles of the jump with the slow axis.

Since $\lambda_{+}\geq\lambda_{-}$, the jump axis is the eigenvector $\vec{V}_{+}$, with a characteristic angle relative to the measurement axis

[TABLE]

This angle characterizes the direction along which the population of the qubit concentrates, and is only defined within the Zeno regime, where dragging occurs. In such regime a qubit state close to a pole of the jump axis eventually jumps to the other pole at a rate $\gamma_{J}=|\lambda_{+}|/2$. Note that for slow dragging velocities, in the limit $\Gamma_{D}\gg 2|\Omega|$, the jump axis aligns with the measurement axis, and the jump rate converges to the familiar form $\Omega^{2}/(2\Gamma_{D})$ Gambetta et al. (2008); Gagen and Milburn (1993); Cook (1988).

Fig. 3a illustrates the jump axis, indicated by a red line, lagging behind the measurement axis at an angle $\theta$. Moreover, Fig. 3b and Fig. 3c show good agreement between theoretical and experimental ensemble dynamics in the frame of the jump axis for dragging velocities in the Zeno regime. We can see the exponentially decaying behaviour in the direction perpendicular to the jump axis, indicating that the population is aligning with it. The lagging angle between the average state and the measurement axis can be understood to arise from competition between the stochastic back-action and rotation.

Without observing the measurement outcome there is an optimal initial measurement axis and rotation velocity that maximizes the fidelity with respect to a target state Shuang et al. (2008). However, as the magnitude of the back-action depends on the measurement outcome, its relative size can be inferred from the measurement record. As a larger positive measurement outcome induces a larger change toward the measurement axis, one can use this effect to post-select on trajectories in which the state was pulled closer to the measurement axis. In Fig. 4, we show fidelity with respect to the target measurement eigenstate for various post-selection criteria. One can see that the more aggressively one post-selects on the integrated voltage, the higher the resulting fidelity. The apparent degradation in fidelity for the most negative postselections is due to insufficient statistics. Thus, measurement allows us not only to drag the state, but also to monitor its dynamics and herald arbitrarily high fidelity. The above dynamics suggest that given a ‘runaway’ state, or an ‘error’, the measurement axis could rotate and drive it back via a feedback protocol, achieving improved control. A feedback protocol achieving such a result has been shown Tanaka and Yamamoto (2012). The idea is to feedback on the measurement axis such that it is always half way between the current state and the target state.

This dynamical control of the measurement operator enables novel capabilities for qubit control, such as the incoherent control demonstrated here, improved incoherent control with feedback Tanaka and Yamamoto (2012), rapid state purification Jacobs (2003); Wiseman and Ralph (2006); Combes et al. (2010), and adaptive measurements Wiseman et al. (2009); Wiseman and Milburn (2009). This measurement scheme also generalizes to multi-level systems. In such multi-level settings, fast measurement rates of certain operators restrict the system to evolve within a particular subspace of the total Hilbert space, which is known as Quantum Zeno Dynamics Facchi and Pascazio (2002); Bretheau et al. (2015); Signoles et al. (2014). Such restriction has been recently shown to enable universal quantum computation within that subspace Burgarth et al. (2014b). Changing these subspaces dynamically through the evolution of the monitored operators is an avenue that has yet to be explored.

Acknowledgments We thank Alexander Korotkov, Andrew Jordan, Juan Atalaya, Emanuel Flurin, Machiel Blok and Howard Wiseman for discussions. L.S.M. acknowledges support from the National Science Foundation (graduate fellowship grant 1106400). and from a Berkeley Fellowship for Graduate Study. This work was supported by the Air Force Office of Scientific Research (grant FA9550-12-1-0378) and the Army Research Office (grant W911NF-15-1-0496).

I Supplemental Material

I.1 Device parameters

The qubit has a transition frequency of $\omega_{q}/2\pi$ = 4.262 GHz, an energy-decay timescale of $T_{1}=60~{}\mu\mathrm{s}$, and a dephasing (Ramsey decay) timescale of $T_{2}^{*}=30~{}\mu\mathrm{s}$. For this experiment we use the second lowest cavity mode, with a frequency of $\omega/2\pi$ = 7.391 GHz, and a linewidth of $\kappa/2\pi$ = 4.3 MHz. The qubit dispersive frequency shift is $\chi/2\pi$ = -0.23 MHz.

I.2 Trajectory reconstruction

The stochastic master equation given in the main text is generated by the following measurement operator

[TABLE]

where $\mathcal{E}_{1-\eta_{i}}$ is a superoperator which models dephasing due to finite quantum efficiency and small additional dephasing taken from the finite measured Rabi time. To ensure positivity of the state when $dt$ is taken to be finite, we use Eq. (5) to numerically propagate the quantum trajectories. The parameters $\Gamma_{D}$ and $\eta$ are calibrated independently. The former we measure by preparing $|+\rangle$ and then performing a Ramsey measurement. We measure the quantum efficiency by preparing states $|y=\pm 1\rangle$. Histograms of the integrated measurement records yield a pair of Gaussians which separate as a function of time. The quantum efficiency is given by

[TABLE]

where $\mu_{y=\pm 1}$ is the mean of the Gaussian for the $|y=\pm 1\rangle$ state preparation, $\sigma$ is the average standard deviation of the Gaussians and $\tau$ is the measurement duration Korotkov (2014, 2016).

When reconstructing the quantum trajectories and comparing the average of the trajectories to the solution for the master equation for the the average state we found a disagreement between theory and experiment. This seemed to be a systematic discrepancy due to a small offset in the detector voltage. We corrected this using an informed ‘guess’ offset, which was calibrated from a different experiment Atalaya et al. (2017) performed using the same setup. The offset value used is 0.17V where the separation of the mean of the Gaussians for this detector was 1.74V. In Fig. 5 we show the comparison of the average of the trajectories with theory for 3 dragging velocities, for processing with and without the correction. In the main text we use the corrected data.