Decoherence of a tunable capacitively shunted flux qubit

Quantum annealing (QA) is a computational paradigm that shows promise for outperforming classical computers in solving hard optimization problems [1, 2, 3]. In comparison to the more commonly pursued gate-model quantum computation (GMQC), QA is more amenable to scaling up in the near term, since it does not require individual dynamic control on each qubit. Outside the context of optimization, QA also offers potential quantum advantages in machine learning and quantum simulation [4, 5]. Moreover, QA is motivated by the fact that the closed-system version of QA, adiabatic quantum computation [6], is polynomially equivalent to GMQC [7, 8].

Superconducting circuits are one of the most prominent physical platforms for both QA and GMQC, owing to a large engineering space and fabrication techniques that build on conventional electronics [9, 10]. In the context of QA, superconducting flux qubits efficiently implement the transverse field Ising Hamiltonian [11, 12, 13], and can be programmed to solve NP-hard optimization problems [14]. Quantum annealers based on this architecture have been made commercially available by D-Wave, with the latest generation devices containing more than 5000 qubits [15]. Recent experiments with these devices demonstrate a scaling advantage over Monte-Carlo methods for solving spin glass problems, showing great potential for quantum-enhanced optimization [16].

Decoherence, or the loss of information from a system to its environment, is an important problem both in GMQC and QA [17, 18, 6]. For a single qubit, decoherence is usually characterized by two time scales, $T_{1}$ and $T_{\phi}$, which correspond to the time it takes for the system to lose energy (relaxation) to the environment, and to lose phase coherence between different eigenstates (dephasing), respectively. In the limit of weak-coupling to an environment, the Bloch-Redfield theory [19, 20] applies and the relaxation and dephasing times are given by [21]

respectively. Here $|{g}\rangle(|{e}\rangle)$ is the qubit ground (excited) state, $H_{q}$ is the qubit Hamiltonian, $\omega_{q}$ is the qubit angular frequency, $\lambda$ is the noise operator, and $S_{\lambda}$ is the noise power spectral density (PSD). The approximate sign in Eq. 2 becomes exact when the noise is white (Markovian limit). It can be seen that decoherence is determined by both the matrix element of the noise coupling operator $\partial H_{q}/\partial\lambda$ and the strength of the noise, characterized by its noise PSD.

Studying and improving the decoherence time of superconducting qubits is critical for building practically relevant quantum computers. The rf-SQUID flux qubit was one of the first superconducting circuits that was identified to behave as a two-state quantum system, i.e., a qubit [22, 23]. Over the last three decades, a few variants of flux qubits have been explored, including the persistent current qubit [24, 25, 26], the capacitively-shunted flux qubits [27, 28, 29, 30], and the fluxonium [31, 32, 33]. A continuous effort to understand the mechanisms of noise in flux qubits has led to an increase of the coherence times from nanoseconds in the first implementations to a millisecond in recent designs [9]. The key to this progress has been the investigation and understanding of a wide range of noise sources, including intrinsic $1/f$ flux noise [34, 35, 36, 37, 38, 39, 40], photon noise in the readout resonator [41, 42, 43, 29, 44], dielectric loss [45, 46, 47, 33, 48, 49, 50] and junction critical-current noise [51, 52]. In spite of the above improvements, the coherence time for the rf-SQUID qubit used in the DWave annealers has been measured to be $\sim 15~{}\text{ns}$ [53], due to a number of reasons, including requirements to simplify control calibration and to allow large coupling strength for annealing applications [11].

The role of decoherence in QA is an open question under active research. Theoretical studies have suggested that in the limit of weak coupling to the environment, decoherence occurs in the instantaneous energy eigenbasis and is not detrimental so long as the system remains in its ground state during the computation [54, 18, 55]. However, in the strong coupling limit, the situation is more complex and decoherence times as given in Eqs. (1,2) are no longer well-defined, as the system and environment become highly correlated. While some previous research suggests that strong coupling to the environment renders coherent quantum annealing impossible [55], other work points out that incoherent quantum-assisted tunneling is still possible, which could still lead to a quantum advantage [56].

Most previous experimental studies of coherence in the context of quantum annealing focused on D-Wave devices [57, 58, 59]. While they offer valuable insight about coherence for a large quantum processor, they often rely on assumptions about the sources of the noise and how strongly coupled they are to the system, which cannot be verified independently. Furthermore, the relatively short coherence time compared to the typically available bandwidth in D-Wave devices (control timescales of the order of $\mu\text{s}$) limits the parameter space that could be explored. That being said, recent work using a D-Wave device on the Kibble-Zurek mechanism when annealing a 2000-qubit spin chain through its paramagnetic to ferromagnetic transition has shown strong agreement with a coherent model when annealing is performed on the time scale of the order of 10’s of ns [60]. This highlights the importance of increasing the control bandwidth relative to the coherence timescale of the qubits, to observe coherent quantum phenomena involving more than just the ground state of many qubits.

In this work, we study the coherence of a single tunable capacitively-shunted flux qubit (CSFQ), designed to be incorporated into a large-scale annealer. The CSFQ is a variant of the persistent-current qubit, and features higher coherence when compared to the rf-SQUID flux qubit used in D-Wave devices, by combining high-quality capacitive shunts and reduced influence of flux noise due to the lower persistent current [29]. We measure the coherence times of the CSFQ for a wide range of flux biases, and model the results with a comprehensive list of noise sources. Although the measured coherence times only apply in the weak-coupling limits, the inferred noise power provides a solid basis for future coherence studies in more complex settings. We found that relaxation is dominated by $1/f$ flux noise up to about $\sim 3~{}\text{GHz}$, with additional contributions from bias line thermal noise and possibly two-level system (TLS) defects [61], while dephasing is dominated by intrinsic flux noise. Additionally we note that electronics required for high bandwidth control introduces challenges related to dephasing. Our results are immediately relevant to upcoming experiments that explore coherent QA based on CSFQs, and could inform the design of other variants of flux qubits, such as the fluxonium.

This article is organized as follows. In Sec. II we introduce the CSFQ, including the control and readout circuitry. This is followed by a discussion of the basic characterization measurement of the CSFQ. In Sec. IV we present the main results of this article, which are the coherence time measurements, and discuss the various noise sources considered. In Sec. V we comment on the effect of flux noise on the parameters used to define an annealing Hamiltonian, followed by conclusions in Sec. VI.

The measured CSFQ, together with the control and readout circuitry is schematically shown in Fig. 1(a). The CSFQ has a main loop and a secondary split-junction loop. In the two-state approximation, the CSFQ has the Hamiltonian

where $\hbar$ is the reduced Planck’s constant, $I_{p}$ is the persistent current in the qubit main loop, $\Delta$ is the tunneling amplitude between the two persistent current states, $\Phi_{z}^{\text{sym}}$ is the effective flux bias symmetry point, and $\sigma_{z,x}$ are the qubit Pauli operators in the persistent current basis. The flux bias $\Phi_{x}$ is the externally applied flux in the secondary loop. The flux bias $\Phi_{z}$ is the effective flux bias in the main loop, defined such that the native crosstalk from the $X$-loop $\Phi_{x}/2$ is accounted for [25]. To a large extent $\Phi_{z}$ controls the longitudinal field strength while $\Phi_{x}$ controls the transverse field strength, hence the name $Z$ and $X$ respectively. The $X$-loop has a symmetrized design, allowing nearly independent control of the qubit’s transverse and longitudinal field (see Appendix B).

The external flux biases are controlled by currents through on-chip flux-bias lines, which are supplied by DC voltage sources and arbitrary waveform generators (AWGs) at room temperature, combined through cryogenic diplexers. In the experiment reported here, the high-frequency port of the diplexer that controls the $X$-loop was unresponsive, and is hence neglected in the schematic in Fig. 1(a). In addition to flux biasing, the CSFQ is capacitively coupled to an rf source, allowing microwave excitation of the qubit.

Readout of the qubit is achieved by coupling the qubit $Z$-loop inductively to an rf-SQUID-terminated $\lambda/4$-resonator, coupled to a transmission line. The external flux bias of the rf-SQUID, $\Phi_{r}$ is controlled by current through the on-chip bias line. When $\Phi_{r}=0$, the resonator is at its maximum frequency and the qubit-resonator coupling is linear. At this point, the resonator experiences a qubit-state-dependent dispersive shift. Away from zero bias, the resonator has a flux-sensitive resonance, which can be used for persistent current basis readout of the qubit (see Appendix C for more details regarding the qubit-resonator interaction).

The device is fabricated at MIT Lincoln Laboratory using a flip-chip process [62], combining a high-coherence qubit chip hosting the CSFQ, and a control chip that hosts the readout and control circuitry. Optical images of the two chips are shown in Fig. 1(b).

In this section, we discuss the characterization measurements for the CSFQ. We note that details of the measurement setup and circuit modelling have been given in a related manuscript [63], and we focus the discussion here on aspects that concern annealing implementation with the CSFQ.

Experiments were performed with the device cooled down in a dilution refrigerator, with base temperature of about 10 mK. The crosstalk between the flux-bias lines was measured using the iterative procedure introduced in Ref. [64]. After three iterations, the error in flux biasing is expected to be about $1~{}\text{m}\Phi_{0}/\Phi_{0}$, with $\Phi_{0}$ being the flux quantum. In Fig. 2 we show the transmission through the line coupled to the resonator, at a frequency near the resonator maximum frequency versus the calibrated external fluxes $\Phi_{z},\Phi_{x}$. This measurement shows that the $\Phi_{z}$ symmetry point has a non-linear dependence on $\Phi_{x}$. This is due to the asymmetry between the two junctions in the DC-SQUID, which results in an effective contribution of the $X$-loop flux to the $Z$-loop [65]. Assuming negligible $X$-loop inductance, the $Z$-loop symmetry point is given by [39]

where $d=(I_{cl}-I_{cr})/(I_{cl}+I_{cr})$ is the junction asymmetry, with $I_{cl(r)}$ being the critical current of the left(right) junction in the DC-SQUID. The symmetry point can be extracted by checking the point of reflection symmetry for each trace of the transmission measurement at each value of $\Phi_{x}$, and the fitted asymmetry is $d=0.069\pm 0.001$. To verify the effect of finite $X$-loop inductance, we use a full circuit model (see Appendix in Ref. [64]) to numerically extract the symmetry points and compare with the analytical expression. As shown in Fig. 2(b), using the same junction asymmetry $d$, the discrepancy between the numerically simulated and analytical $\Phi_{z}^{\text{sym}}$ grows larger as $\Phi_{x}$ increases, but always remain below $1.5~{}\text{m}\Phi_{0}$ for the range where the coherence measurements were performed, justifying the use of the analytical expression.

Once the crosstalk and junction asymmetry are calibrated, spectroscopy of the qubit is performed at a range of $\Phi_{z}$ and $\Phi_{x}$ biases, with the resonator SQUID biased at $\Phi_{r}=0$. A circuit model of the device is fit to the transition frequencies between ground, first and second excited states. The model includes the full capacitance matrix between islands, the loop inductances, and the Josephson junction critical currents, and is simulated using a package developed in Ref. [66] (see Appendix in Ref. [64] for circuit parameters).

Using the circuit model, the qubit persistent current $I_{p}$ and tunneling amplitude $\Delta$ versus the flux bias in the $X$-loop $\Phi_{x}$ can be obtained. The persistent current $I_{p}$ can be calculated using two different methods. In the first method, we rely on the operator matrix element defined on the CSFQ circuit

where $|{0_{\text{c}}}\rangle(|{1_{\text{c}}}\rangle)$ is the circuit ground (excited) state, and $H_{\text{c}}$ is the circuit Hamiltonian, all evaluated at the symmetry point $\Phi_{z}^{\text{sym}}$. In the second method, we find the ground-to-excited-states transition frequencies and fit to the qubit frequencies given by the two-state Hamiltonian Eq. 3. The corresponding persistent current is denoted as $I_{p,\text{q}}$. In Fig. 3 we show $I_{p,\text{c}},I_{p,\text{q}}$ versus $\Phi_{x}$. It can be seen that they agree with each other for small values of $\Phi_{x}$. At larger $\Phi_{x}$, $I_{p,\text{q}}$ becomes smaller than $I_{p,\text{c}}$. This can be seen as a result of the breakdown of the two-state approximation, as the lowest energy states of the circuit are no longer spanned by the two persistent current states as $\Phi_{x}$ (or equivalently the tunneling amplitude $\Delta$) increases. This could lead to deviation from the transverse field Ising model typically used to describe a quantum annealer, though its impact is likely small for adiabatic evolution. Its impact in the case of non-adiabatic evolution will be studied in future work.

When compared with rf-SQUID flux qubits used in commercially available annealers, the CSFQ presented here has a persistent current that is at least an order of magnitude smaller, hence reducing its sensitivity to flux noise. The smaller $I_{p}$ also leads to a smaller interaction strength between qubits, which potentially limits the optimization problems that can be mapped to the annealer. We note that this could be compensated by galvanic coupling, using shared segments of loops [67], and even shared junction or junction arrays between the circuits that need to be coupled.

The simulated tunneling amplitude $\Delta$, also the qubit frequency at the symmetry point, is shown in Fig. 3. To implement a standard annealing experiment, the CSFQ is initialized at a large $\Delta$, around $\Phi_{x}\approx 0.5$, and then the tunneling amplitude is gradually reduced following a specific annealing schedule. The CSFQ has a more gentle dependence of $\Delta$ on $\Phi_{x}$ as compared to the rf-SQUID qubit. Given the same resolution in the biasing sources, this $\Delta$ dependence allows finer control of the annealing schedule, and assists in future experimental demonstrations of novel annealing protocols, such as those proposed in Ref. [68, 69].

In this section we present the measurements of the relaxation and dephasing times $T_{1},T_{\phi}$ of the CSFQ. We note that $T_{1},T_{\phi}$ coherence times are only well-defined when the system is weakly coupled to the environment, which is commonly the case in devices made for GMQC. In the context of annealing, the validity of the weak-coupling approximation depends on the relative strength of the qubit fields and the noise. In this work, we measure the coherence times for qubit frequency down to $\Delta/(2\pi)\approx 1\text{GHz}$, below which thermally excited qubit population becomes significant. Although qubit state initialization at lower qubit frequencies is possible through high-fidelity single-shot readout [70], we do not pursue them in this work, as the available measurement range is enough to determine the strengths of the different noise, and for small enough $\Delta$, the weak-coupling limit eventually breaks down [63].

It is useful to put the noise measured here in the context of annealing applications. For a “single-qubit anneal” [65, 78], the qubit Hamiltonian starts with a large value of $\Delta$, at which the qubit is initialized in the ground state, and the qubit Hamiltonian is gradually changed to a target Hamiltonian where $\Delta$ is close to zero. The only parameter defining the target Hamiltonian is the qubit longitudinal field

Noise in the control fluxes gives rise to noise in the annealing parameters $\epsilon,\Delta$, with the same frequency dependence. We focus on the intrinsic $1/f$ flux noise, assuming noise arising from wiring and electronics can be minimized through careful engineering. Using the two-state approximation Hamiltonian Eq. 3, we find the individual and correlated $1/f$ noise in $\epsilon,\Delta$ to be

where $A_{\Phi_{z}\Phi_{x}}=c_{zx}\sqrt{A_{\Phi_{z}}A_{\Phi_{x}}}$ is the $1/f$ power of the flux noise correlation. In Fig. 6 we show these three quantities versus the transverse field $\Delta$, for the target longitudinal field $\epsilon=0$, noting these quantities are weakly dependent on $\epsilon$. It can be seen that the $\epsilon$ noise, $A_{\epsilon}$ is always the dominant noise factor. The correlated noise $A_{\Delta\epsilon}$ is mainly due to the measured flux noise correlation, and is about an order of magnitude lower than $A_{\epsilon}$. The noise in $A_{\Delta}$ is comparable to $A_{\Delta\epsilon}$ for large $\Delta$, but diminishes quickly with decreasing $\Delta$. The role of the relative strength of the noises will be explored in future work. One interesting direction is to extend the previous analysis of the effect of correlated noise on single qubit Landau-Zener tunneling [79], to the general case of multi-qubit annealing.

We have presented a detailed characterization of coherence in a CSFQ design relevant for quantum annealing. We measured the $T_{1}$ relaxation and $T_{\phi}$ dephasing times and modelled them considering all relevant noise sources. We find that the $T_{1}$ values are influenced primarily by $1/f$ flux noise with additional contributions from bias line thermal noise and TLSs at higher frequencies. The measured dephasing time $T_{\phi}$ is dominated by $1/f$ flux noise, with a possible contribution from $1/f$ noise in the junction critical current. We point out that for annealing applications, the external flux biases can introduce significant relaxation and dephasing as compared to the intrinsic noise of the qubit. This suggests that future annealing experiments that aim to combine high-coherence and high control flexibility need to carefully evaluate the trade-off between the added noise and the improved control bandwidth.

Additionally, we observe a positive correlation between the $Z$ and $X$ flux noise, as manifested in the maximum dephasing time versus $\Phi_{z}$ occurring to the left of the $Z$ bias symmetry point. This correlation may arise due to environmental spins on the interposer ground plane, and junction critical-current noise. Future experiments that are able to distinguish the different sources of $1/f$ noise are needed to quantitatively determine the origin of this noise correlation.

Our work provides a detailed characterization of noise sources of a flux qubit, relevant for future analysis of coherent annealers based on CSFQs. Besides quantum annealing, the work presented here is also relevant for flux qubit devices made for GMQC, such as fluxonium qubits. Future work will be directed at extending the coherence characterization to the strong coupling limit and multi-qubit systems, as well as studying coherence in a dynamic setting, which will lead to a deeper understanding of the role of coherence in quantum annealing.