Fast and High-Fidelity Entangling Gate through Parametrically Modulated Longitudinal Coupling
We investigate an approach to universal quantum computation based on the modulation of longitudinal qubit-oscillator coupling. We show how to realize a controlled-phase gate by simultaneously modulating the longitudinal coupling of two qubits to a common oscillator mode. In contrast to the more familiar transversal qubit-oscillator coupling, the magnitude of the effective qubit-qubit interaction does not rely on a small perturbative parameter. As a result, this effective interaction strength can be made large, leading to short gate times and high gate fidelities. We moreover show how the gate infidelity can be exponentially suppressed with squeezing and how the entangling gate can be generalized to qubits coupled to separate oscillators. Our proposal can be realized in multiple physical platforms for quantum computing, including superconducting and spin qubits.
Current address: Rigetti Quantum Computing, 775 Heinz Avenue, Berkeley, California 94710, USA.
Introduction—A widespread strategy for quantum information processing is to couple the dipole moment of multiple qubits to common oscillator modes, the latter being used to measure the qubits and to mediate long-range interactions. Realizations of this idea are found in Rydberg atoms Haroche and Raimond (2006), superconducting qubits Blais et al. (2004) and quantum dots Imamoglu et al. (1999) amongst others. With the dipole moment operator being off-diagonal in the qubit’s eigenbasis, this type of transversal qubit-oscillator coupling leads to hybridization of the qubit and oscillator degrees of freedom. In turn, this results in qubit Purcell decay Houck et al. (2008) and to qubit readout that is not truly quantum non-demolition (QND) Boissonneault et al. (2009). To minimize these problems, the qubit can be operated at a frequency detuning from the oscillator that is large with respect to the transverse coupling strength . This interaction then only acts perturbatively, taking a dispersive character Haroche and Raimond (2006). While it has advantages, this perturbative character also results in slow oscillator-mediated qubit entangling gates Blais et al. (2007); Chow et al. (2013); Paik et al. (2016).
Rather than relying on the standard transversal coupling, , an alternative approach is to use a longitudinal interaction, Kerman and Oliver (2008); Kerman (2013); Billangeon et al. (2015a, b); Didier et al. (2015); Richer and DiVincenzo (2016). Since commutes with the qubit’s bare Hamiltonian the qubit is not dressed by the oscillator. Purcell decay is therefore absent Kerman (2013); Billangeon et al. (2015a) and qubit readout is truly QND Didier et al. (2015). The absence of qubit dressing also allows for scaling up to a lattice of arbitrary size with strictly local interactions Billangeon et al. (2015a).
By itself, longitudinal interaction however only leads to a vanishingly small qubit state-dependent displacement of the oscillator field of amplitude , with the oscillator frequency. In Ref. Didier et al. (2015), it was shown that modulating at the oscillator frequency activates this interaction leading to a large qubit state-dependent oscillator displacement and to fast QND qubit readout. In this paper, we show how the same approach can be used, together with single qubit rotations, for universal quantum computing by introducing a fast and high-fidelity controlled-phase gate based on longitudinal coupling. The two-qubit logical operation relies on parametric modulation of a longitudinal qubit-oscillator coupling, inducing an effective interaction between qubits coupled to the same mode. A similar gate was first studied in Ref. Kerman (2013) in the presence of an additional dispersive interaction and a cavity drive. We show that, with a purely longitudinal interaction excluding the former term, the gate fidelity can be improved exponentially using squeezing. We moreover show that the gate can be performed remotely on qubits coupled to separate but interacting oscillators. The latter allows for a modular architecture that relaxes design constraints and avoids spurious interactions while maintaining minimal circuit complexity Billangeon et al. (2015a); Richer and DiVincenzo (2016); Brecht et al. (2016).
In contrast to two-qubit gates based on a transversal interaction Blais et al. (2007); Chow et al. (2013), this proposal does not rely on strong qubit-oscillator detuning and is not based on a perturbative argument. As a result, the longitudinally mediated interaction is valid for all qubit, oscillator and modulation parameters and does not result in unwanted residual terms in the Hamiltonian. For this reason, in the ideal case where the interaction is purely longitudinal (i.e. described by ), there are no fundamental bounds on gate infidelity or gate time and both can in principle be made arbitrarily small simultaneously.
Similarly to other oscillator-mediated gates, loss from the oscillator during the gate leads to gate infidelity. This can be minimized by working with high-Q oscillators something that is, however, in contradiction with the requirements for fast qubit readout Blais et al. (2004). We solve this dilemma by exploiting quantum bath engineering, using squeezing at the oscillator input. By appropriately choosing the squeezed quadrature, we show how ‘which-qubit-state’ information carried by the photons leaving the oscillator can be erased. This leads to an exponential improvement in gate fidelity with squeezing strength.
Oscillator mediated qubit-qubit interaction—Following Ref. Didier et al. (2015), we consider two qubits coupled to a single harmonic mode via their degree of freedom. Allowing for a time-dependent coupling, the Hamiltonian reads ()
In this expression, and are the frequencies of the oscillator and of the qubit, respectively, while are the corresponding longitudinal coupling strengths.
For constant couplings, , the longitudinal interaction only leads to a displacement of order , which is vanishingly small for typical parameters. This interaction can be rendered resonant by modulating at the oscillator frequency leading to a large qubit-state dependent displacement of the oscillator state. Measurement of the oscillator by homodyne detection can then be used for fast QND qubit readout Didier et al. (2015). Consequently, modulating the coupling at the oscillator frequency rapidly dephases the qubits. To keep dephasing to a minimum, we instead use an off-resonant modulation of at a frequency detuned from by many oscillator linewidths : , where are constant real amplitudes Kerman (2013).
The oscillator-mediated qubit-qubit interaction can be made more apparent by applying a polaron transformation with an appropriate choice of (see supplemental material). Doing this, we find in the polaron frame the simple Hamiltonian
The full expression for the -coupling strength is given in the supplemental material. In the following we will, however, assume two conditions on the total gate time, , such that this expression simplifies greatly. For and , with and integers, we can replace by
where is the modulation drive detuning.
By modulating the coupling for a time , evolution under Eq. (2) followed by single qubit -rotations leads to the entangling controlled-phase gate . Since together with single qubit rotations forms a universal set Nielsen and Chuang (2000), we only consider this gate from now on.
Note that the conditions on the gate time used in Eq. 3 are not necessary for the validity of Eq. 2, and the gate can be realized without these assumptions. However, as we will discuss below, these conditions are important for optimal gate performance: They ensure that the oscillator starts and ends in the vacuum state, which implies that the gate does not need to be performed adiabatically. Finally, not imposing the second constraint, , only introduces fast rotating terms to Eq. 3 which we find to have negligible effect for the parameters used later in this paper. In other words, this constraint can be ignored under a rotating-wave approximation.
The above situation superficially looks similar to controlled-phase gates based on transversal coupling and strong oscillator driving Blais et al. (2007); Cross and Gambetta (2015); Puri and Blais (2016). There are, however, several key differences. With transversal coupling, the interaction is derived using perturbation theory and is thus only approximately valid for small , with the qubit-oscillator detuning and the oscillator-drive detuning. For the same reason, it is also only valid for small photon numbers Blais et al. (2007). Moreover, this interaction is the result of a fourth order process in , leading to slow gates. Because of the breakdown of the dispersive approximation, attempts to speed up the gate by decreasing the detunings or increasing the drive amplitude have resulted in low gate fidelities Chow et al. (2013). In contrast, with longitudinal coupling, the interaction conveniently scales as , i.e. it scales as a second-order process in , but the exact nature of the transformation means that there are no higher order terms. Consequently, Eq. 2 is valid for any value of , independent of the oscillator photon number. As will become clear later, this implies that the gate time and the gate infidelity can be decreased simultaneously. Finally, with longitudinal coupling, there is no constraint on the qubit frequencies, in contrast with usual oscillator-induced phase gates where the detuning between qubits should preferably be small.
Oscillator-induced qubit dephasing—Fig. 1 illustrates, for , the mechanism responsible for the qubit-qubit interaction. Under longitudinal coupling, the oscillator field is displaced in a qubit-state dependent way, following the dashed lines in Fig. 1(a) (Panels (b) and (c) will be discussed later). This conditional displacement leads to a non-trivial qubit phase accumulation. This schematic illustration also emphasizes the main cause of gate infidelity for this type of controlled-phase gate, irrespective of its longitudinal or transversal nature: Photons leaking out from the oscillator during the gate carry information about the qubit state, leading to dephasing.
A quantitative understanding of the gate infidelity under photon loss can be obtained by deriving a master equation for the joint qubit-oscillator system. While general expressions are given in the supplemental material, to simplify the discussion we assume here that . Following the standard approach Gardiner and Zoller (2004), the Lindblad master equation in the polaron frame reads
where is photon decay rate and denotes the usual dissipation super-operator . The last term of Eq. 4 corresponds to a dephasing channel with rate . Since does not generate qubit-oscillator entanglement during the evolution, we can ensure that in this frame, gate-induced dephasing only happens due to the last term in Eq. 4 with rate , by imposing that the initial and final polaron transformations also do not lead to qubit-oscillator entanglement. This translates to the condition and is realized for , which is the constraint mentioned earlier (neglecting fast-rotating terms related to the second constraint ). More intuitively, it amounts to completing full circles in Fig. 1, the oscillator ending back in its initial unentangled state. Note that these conclusions remain unchanged if the oscillator is initially in a coherent state. As a result, there is no need for the oscillator to be empty at the start of the gate Kerman (2013).
Based on the dephasing rate and on the gate time , a simple estimate for the scaling of the gate infidelity is 111Note that refers only to the error due to photon decay, excluding the qubits natural and times.. A key observation is that this gate error is independent of , while the gate time scales as . Both the gate time and the error can therefore, in principle, be made arbitrarily small simultaneously. This scaling of the gate error and gate time is confirmed by the numerical simulations of Fig. 2, which shows the dependence of the gate infidelity Nielsen (2002) on detuning and coupling strength , as obtained from numerical integration of Eq. 4. The expected increase in both fidelity (full lines) and gate time (dashed lines) with increasing detuning are apparent in panel (a). In addition, panel (b) confirms that, to a very good approximation, the fidelity is independent of (full lines) while the gate time decreases as (dashed lines).
This oscillator-induced phase gate can be realized in a wide range of physical platforms where longitudinal coupling is possible. Examples include spin qubits in inhomogeneous magnetic field Beaudoin et al. (2016), singlet-triplet spin qubits Jin et al. (2012), flux qubits capacitively coupled to a resonator Billangeon et al. (2015a) and transmon-based superconducting qubits Didier et al. (2015); Richer and DiVincenzo (2016); Kerman (2013). The parameters used in Fig. 2 have been chosen following the latter references. In particular, taking MHz Bruno et al. (2015), MHz Didier et al. (2015) and MHz results in a very short gate time of 37 ns with an average gate infidelity as small as 1 . Taking into account finite qubit lifetimes s and s Corcoles et al. (2015), we find that the infidelity is increased to (see supplemental material). In other words, the gate fidelity is limited by the qubit’s natural decoherence channels with these parameters. For a comparison with transversal resonator-induced phase gate, see the supplemental material.
A crucial feature of this gate is that the circular path followed by the oscillator field in phase space maximizes qubit-state dependent phase accumulation while minimizing dephasing, allowing for high gate fidelities. In contrast to Kerman (2013), this relies on the assumption that there is no dispersive interaction of the form in Eq. 1. Furthermore, we show below that this also allows for exponential improvement in gate fidelity with squeezing. It is therefore desirable to minimize, or avoid completely, dispersive coupling in experimental implementations 222In the proposal of Ref. Didier et al. (2015), this can be done by reducing the participation ratio, , such that .
Improved fidelity with squeezing—
As discussed above, for fixed and the fidelity increases with decreasing . A small oscillator decay rate , however, comes at the price of longer measurement time if the same oscillator is to be used for readout Didier et al. (2015). This problem can be solved by sending squeezed radiation to the oscillator’s readout port. As schematically illustrated in Fig. 1, by orienting the squeezing axis with the direction of the qubit-dependent displacement of the oscillator state, the which-path information carried by the photons leaving the oscillator can be erased. By carefully choosing the squeezing angle and frequency, it is thus possible to improve the gate performance without reducing . We now show two different approaches to realize this, referring the reader to the supplemental material for technical details.
A first approach is to send broadband two-mode squeezed vacuum at the input of the oscillator, where the squeezing source is defined by a pump frequency and a squeezing spectrum with large degree of squeezing at and . A promising source of this type of squeezing is the recently developed Josephson travelling wave amplifiers Macklin et al. (2015); White et al. (2015). With such a squeezed input field, a coherent state of the oscillator becomes a squeezed state with a squeezing angle that rotates at a frequency . As illustrated in Fig. 1(b), this is precisely the situation where the anti-squeezed quadrature and the displacement of the oscillator’s state are aligned at all times. This leads to an exponential decrease in dephasing rate
with the squeezing parameter. This reduction in dephasing rate leads to the exponential improvement in gate fidelity with squeezing power shown by the brown line in Fig. 3(c). An interesting feature in this Figure is that increasing by 2 orders of magnitude to allow for fast measurement Didier et al. (2015), leads to the same gate infidelity obtained above without squeezing here using only 6 dB of squeezing. Since numerical simulations are intractable for large amount of squeezing, we depict the infidelity obtained from a master equation simulation by a solid line and the expected infidelity from analytical calculations by a dash-dotted line.
An alternative solution is to use broadband squeezing centered at the oscillator’s frequency, i.e. a squeezing source defined by a pump frequency . As illustrated in Fig. 1(c), using this type of input leads to a squeezing angle that is constant in time in a frame rotating at . With this choice, information about the qubits’ state contained in the quadrature of the field is erased while information in the quadrature is amplified (cf. Fig. 1). By itself, this does not lead to a substantial fidelity improvement. However, a careful treatment of the master equation shows that Eq. 5 can be recovered by adding a filter reducing the density of modes at to zero at the output port of the oscillator (see supplemental material). Filters of this type are routinely used experimentally to reduce Purcell decay of superconducting qubits Reed et al. (2010); Bronn et al. (2015). As illustrated by the dark blue line in Fig. 3(c), using single-mode squeezing at and a filter at the modulation frequency, we recover the same exponential improvement found with two-mode squeezing, Eq. 5, in addition to a factor of two decrease in gate infidelity without squeezing.
Interestingly, rotating the squeezing axis by when squeezing at the oscillator frequency helps in distinguishing the different oscillator states and has been shown to lead to an exponential increase in the signal-to-noise ratio for qubit readout Didier et al. (2015). In practice, the difference between performing a two-qubit gate and a measurement is thus the parametric modulation frequency (off-resonant for the gate and on resonance for measurement) and the choice of squeezing axis.
We note that Eq. 5 was derived from a master equation treatment under the standard secular approximation Breuer and Petruccione (2007), which is not valid at high squeezing powers (here, dB, see supplemental material). At such high powers, the frequency dependence of together with other imperfections are likely to be relevant.
Scalability—So far we have focused on two qubits coupled to a single common oscillator. As shown by Billangeon et al. Billangeon et al. (2015a), longitudinal coupling of several qubits to separate oscillators that are themselves coupled transversely has favorable scaling properties. Circuits implementing this idea were also proposed by Richer et al. Richer and DiVincenzo (2016). Interestingly, the gate introduced in this paper can also be implemented in such an architecture. Consider two qubits interacting with distinct, but coupled, oscillators with the corresponding Hamiltonian Billangeon et al. (2015a)
In this expression, , label the mode of each oscillator of respective frequencies , and is the oscillator-oscillator coupling. As above, are modulated at the same frequency , corresponding to the detunings and . Following the same procedure as above and performing a rotating-wave approximation for simplicity, we find a Hamiltonian in the polaron frame of the same form as Eq. 2, but now with a modified interaction strength
where and . This implementation allows for a modular architecture, where each unit cell is composed of a qubit and coupling oscillators, used for both readout and entangling gates. Such a modular approach can relax design constraints and avoids spurious interactions with minimal circuit complexity Billangeon et al. (2015a); Richer and DiVincenzo (2016); Brecht et al. (2016).
Conclusion—We have proposed a controlled-phase gate based on purely longitudinal coupling of two qubits to a common oscillator mode. The key to activating the qubit-qubit interaction is a parametric modulation of the qubit-oscillator coupling at a frequency far detuned from the oscillator. The gate infidelity and gate time can in principle be made arbitrarily small simultaneously, in stark contrast to the situation with transversal coupling. We have also shown how the gate fidelity can be exponentially increased using squeezing and that it is independent of qubit frequencies. The gate can moreover be performed remotely in a modular architecture based on qubits coupled to separate oscillators. Together with the fast, QND and high-fidelity measurement scheme presented in Ref. Didier et al. (2015), this makes a platform based on parametric modulation of longitudinal coupling a promising path towards universal quantum computing in a wide variety of physical realizations.
Acknowledgements.We thank J. Bourassa, D. Poulin and S. Puri for useful discussions. This work was supported by the Army Research Office under Grant No. W911NF-14-1-0078 and NSERC. This research was undertaken thanks in part to funding from the Canada First Research Excellence Fund and the Vanier Canada Graduate Scholarships.
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