Analytic Design of Accelerated Adiabatic Gates in Realistic Qubits: General Theory and Applications to Superconducting Circuits
F. Setiawan, Peter Groszkowski, Hugo Ribeiro, Aashish A. Clerk
SSuperconducting qubit gates via analytically-derived accelerated adiabatic pulses
F. Setiawan, ∗ Peter Groszkowski, Hugo Ribeiro, and Aashish A. Clerk Pritzker School of Molecular Engineering, University of Chicago,5640 South Ellis Avenue, Chicago, Illinois 60637, USA Max Planck Institute for the Science of Light, Staudtstraße 2, 91058 Erlangen, Germany (Dated: February 5, 2021)Shortcuts to adiabaticity (STA) is a general methodology for speeding up adiabatic quantumprotocols, and has many potential applications in quantum information processing. Unfortunately,analytically constructing STAs for systems having complex interactions and more than a few levelsis a challenging task. This is usually overcome by assuming an idealized Hamiltonian (e.g., only alimited subset of energy levels are retained, and the rotating-wave approximation (RWA) is made).Here, we develop an analytic approach that allows one to go beyond these limitations. Our methodis general and results in analytically-derived pulse shapes that correct both non-adiabatic errorsas well as non-RWA errors. We also show that our approach can yield pulses requiring a smallerdriving power than conventional non-adiabatic protocols. We show in detail how our ideas can beused to analytically design high-fidelity single-qubit “tripod” gates in a realistic superconductingfluxonium qubit.
I. INTRODUCTION
Quantum gates based on adiabatic evolution [1–7] aregenerally desirable because of their intrinsic robustnessagainst imperfections in control pulses, and have beenimplemented in a variety of platforms (see e.g., [8–10]).They, however, require extremely long evolution times,making them potentially susceptible to dissipation andnoise. An intriguing possibility is to try to “accelerate”adiabatic gates using techniques drawn from the field ofshortcuts to adiabaticity (STA) [11–15]. STA protocolsseek to modify pulses to completely cancel non-adiabaticerrors. They are usually developed for simple evolutionsthat do not correspond to a true quantum gate, as theirform is tied to a specific choice of initial state. However,they can be adapted for true gates. Ref. [16] showedhow a particular shortcut approach (superadiabatic tran-sitionless driving (SATD) [17]) could be used to acceler-ate a true (arbitrary) single-qubit gate, the paradigmatic“tripod” adiabatic gate introduced in Refs. [3, 4] (seeFig. 1). Other STA approaches to gates have been pre-sented in Refs. [18–22]. Note that STA-accelerated gatesare conceptually distinct from the so-called non-adiabaticholonomic gates (see e.g., Ref. [23]), and have distinct ad-vantages [16].While the above results are promising, they are lim-ited by a constraint that plagues most STA approachesto quantum control: accelerated protocols can only be de-rived analytically for few-level systems (or systems thatreduce to uncoupled few-level systems). Further, oneneeds to ignore fast-oscillating non-resonant terms (i.e.,one must necessarily make the rotating-wave approxima-tion (RWA)). In many realistic settings, unwanted non-RWA dynamics and couplings to higher levels cannot beneglected, and will limit the operation fidelity even if ∗ [email protected] − − − . − . − . . . . . E n e r g y ( G H z ) | e (cid:105) | a (cid:105)| (cid:105) | (cid:105) phase ϕ/ π FIG. 1. A single-qubit gate realized using a 4-level tripod en-ergy structure consisting of three lower energy levels ( | (cid:105) , | (cid:105) and | a (cid:105) ) resonantly coupled to an excited state | e (cid:105) by threedifferent driving tones: each with envelope Ω ( t ), Ω ( t ), andΩ ae ( t ), respectively. This tripod-energy structure can be engi-neered in a multi-energy spectrum (shown above are 9 lowestenergy levels) of a fluxonium qubit whose device schematicis shown in the inset (upper left corner). The driving fieldscan drive unwanted transitions which give rise to coherent er-rors, e.g., transitions between the energy levels outside thetripod structure (shown using light gray) and the energy lev-els in the tripod structure as well as the transitions inside thetripod structure driven by non-resonant tones. non-adiabatic errors are suppressed. For this reason, theutility of analytic STA protocols for high-fidelity opera-tions in complex systems have remained unclear.In this paper, we present a generic approach for im-proving STA protocols in settings where assuming anidealized dynamics is not possible, e.g., non-RWA termscannot be neglected. The result is a general method for analytically deriving pulse sequences that both fully can-cel non-adiabatic errors, and partially mitigate non-RWA a r X i v : . [ qu a n t - ph ] F e b errors. To highlight the efficacy of our approach, we focuson a specific, experimentally relevant setting: an acceler-ated geometric tripod gate implemented in a fluxonium-style superconducting circuit [24–27]. The isolated qubitlevels of this system make conventional approaches togates problematic, motivating new ideas that do not re-quire a direct coupling of qubit levels. The so-called tri-pod gate [3, 4] is a natural candidate. However, as shownin Fig. 1, a fluxonium circuit has a complex level struc-ture, implying that non-resonant, non-RWA correctionswill be important. Using realistic parameters compatiblewith experiment, we show (via full master-equation solu-tions including dissipation, non-RWA effects, and powerconstraints from cavity-based driving) that our enhancedSTA gate achieves gate fidelities of ≈ . t g = 100 ns. This is roughly comparable to resultsobtained using fully numerical optimal control on a re-lated superconducting circuit [28]. It also demonstratesthat the accelerated tripod-gate can be advantageous, de-spite the ability to realize a perfectly isolated tripod levelstructure. Note that the use of our enhanced protocol iscrucial: if one simply uses the STA derived without cor-rections, the fidelity error for the same gate time is ordersof magnitude worse.The paper is organized as follows. We begin in Sec. IIby introducing the most general version of our problem:how can one analytically design STA protocols in com-plex multi-level systems? In Sec. III, we briefly reviewthe basic (RWA) geometric tripod gate [3, 4] as well asits accelerated version [16]. In Sec. IV, we go beyondthe RWA, and discuss how in general STA approachescan be further enhanced to mitigate non-resonant errors.In Sec. V, we explore the utility of these methods byapplying them to design an accelerated gate in a realis-tic fluxonium superconducting circuit. Results for gateperformance are presented in Sec. VI, and comparisonagainst a simpler “direct-driving” gate is presented inSec. VII. We summarize our results in Sec. VIII. II. GENERAL PROBLEM:SHORTCUT-TO-ADIABATICITY APPROACHESFOR COMPLEX DRIVEN SYSTEMS
We begin by considering a generic driven multilevelsystem whose Hamiltonian in the lab frame has the formˆ H ( t ) = (cid:88) k ε k | k (cid:105)(cid:104) k | + V ( t ) (cid:88) ( k,l ) | l>k n kl | k (cid:105)(cid:104) l | + H . c . , (1)where ε k , | k (cid:105) are the eigenenergies and eigenstates of theundriven system, and n kl = (cid:104) k | ˆ n | l (cid:105) is an effective dipolematrix element. The full control pulse V ( t ) consists ofseveral distinct drive tones ω j , each associated with a slowly-varying complex envelope V j ( t ), i.e., V ( t ) = 12 (cid:88) j (cid:0) V j ( t ) e iω j t + H.c. (cid:1) . (2)We next move to an interaction picture defined byˆ U diag = e − i ˆ H diag t where ˆ H diag = (cid:80) k ε k | k (cid:105)(cid:104) k | . TheHamiltonian in this frame takes the general formˆ H ( t ) = ˆ H ( t ) + ˆ H err ( t ) . (3)Here ˆ H ( t ) describes resonant processes, and is time-dependent only through its dependence on the envelopefunctions V j ( t ). Defining ε kl = ε l − ε k , we haveˆ H ( t ) = 12 (cid:88) j (cid:88) ( k,l ) | l>k & ε kl = ω j (cid:16) V j ( t ) n kl | k (cid:105)(cid:104) l | + H . c . (cid:17) . (4)In contrast, ˆ H err ( t ) describes all non-resonant processesˆ H err ( t ) = 12 (cid:32) (cid:88) j (cid:88) ( k,l ) | l>k & ε kl (cid:54) = ω j V j ( t ) e − i ( ε kl − ω j ) t n kl | k (cid:105)(cid:104) l | + (cid:88) j (cid:88) ( k,l ) | l>k & ε kl (cid:54) = − ω j V ∗ j ( t ) e − i ( ε kl + ω j ) t n kl | k (cid:105)(cid:104) l | (cid:33) + H . c . (5)The standard next step in most analytic STA ap-proaches to quantum control is to make the rotatingwave approximation (RWA): one assumes that the en-ergy detunings in ˆ H err ( t ) are sufficiently large that thiserror Hamiltonian can be approximated as zero. The re-sult is a much simpler Hamiltonian that only involvesthe slowly varying amplitudes V j ( t ), and which typicallyonly couples a small subset of levels. It is in this contextthat many exact STA protocols have been derived; theseprotocols yield a perfect, error-free evolution within theRWA. Examples range from accelerated versions of thetwo-level Landau-Zener problem [11, 13, 14, 29], to morecomplex three-[17, 30–33] and four-[16] level protocols.Despite the power of the above STA approaches, theyonly address non-adiabatic errors associated with theRWA Hamiltonian ˆ H ( t ). A crucial question is whetherthey can also be adapted to address additional errors aris-ing from the non-RWA dynamics described by ˆ H err ( t ).Corrections to the RWA are important in many physicalsystems if one is interested in realizing truly high-fidelityoperations. One could in principle try to derive an STAfor the full multilevel fast-oscillating Hamiltonian H ( t ).But in most cases, this is completely infeasible. Notonly does this involve dealing with a large-dimensionalHilbert space, but it also involves working with a start-ing Hamiltonian that has extremely fast time-dependentterms (i.e., in ˆ H err ( t )), and hence is nowhere close to anadiabatic limit.A central goal of this work is to present a much moretractable approach to adapting exact STA protocols sothat they also mitigate non-resonant, non-RWA errors.Our method is ultimately perturbative, and amounts tomodifying the original (RWA) STA protocol to correctthe leading effects of ˆ H err ( t ). We stress that our approachretains the crucial feature of the original STA pulse se-quence of being described and derived fully analytically,i.e., no recourse is made to numerical optimal-control ap-proaches. While our method is extremely general, we willfocus in what follows on a particularly promising proto-col involving accelerated geometric gates using a tripodlevel structure [3, 4] [see Fig. 2(a)]. Note that the generalapproach we present is completely distinct from recentworks [34, 35] which deliberately introduce additionalhigh-frequency oscillatory terms to a RWA Hamiltonianto approximately engineer desired STA protocols. III. REVIEW: ACCELERATED ADIABATICQUANTUM GATES
In this section we briefly review the basic geometrictripod gate introduced in Refs. [3, 4], and its acceleratedversion [16]. All these analyses were done in the contextof a simplified four-level RWA Hamiltonian. Our reviewhere will set the stage for our following discussion on howthese approaches can be modified and effectively imple-mented in a realistic multi-level superconducting circuitwhere non-RWA effects play a crucial role.
A. Double STIRAP protocol in an ideal tripodsystem
Starting with the full driven Hamiltonian in Eq. (1),we assume a situation where within the RWA, we realizea so-called tripod level configuration [see Fig. 2(a)]. Anideal tripod system consists of three lower levels (labeledby | (cid:105) , | (cid:105) and | a (cid:105) ) which are controllably coupled to acommon excited state | e (cid:105) [see Fig. 2(a)]. Denoting these(complex) couplings as Ω j e ( t ) ( j = 0 , , a), the tripodHamiltonian is ( (cid:126) = 1):ˆ H ( t ) = 12 [Ω ( t ) | (cid:105)(cid:104) e | + Ω ( t ) | (cid:105)(cid:104) e | + Ω ae ( t ) | a (cid:105)(cid:104) e | + H . c . ] , (6)where Ω j e ( t ) = V j e ( t ) n j e , (7)for j = 0 , , a. We will take the states | (cid:105) and | (cid:105) toencode a logical qubit, while | a (cid:105) and | e (cid:105) serve as auxiliarystates used to perform gate operations.The basic idea of the gate is that ˆ H ( t ) always has twodegenerate zero-energy adiabatic eigenstates, and hencecyclic adiabatic evolution can result in a non-trivial geo-metric 2 × (a) (b)Tripod Λ systemqubit qubit FIG. 2. (a) An ideal tripod system consisting of three lowerenergy levels (the qubit states | (cid:105) and | (cid:105) as well as the aux-iliary state | a (cid:105) ) resonantly coupled to an excited state | e (cid:105) by three different driving tones: each with envelope Ω ( t ),Ω ( t ), Ω ae ( t ), respectively. (b) An effective Λ system (con-sisting of the state | ˜1 (cid:105) and | a (cid:105) resonantly coupled to the state | e (cid:105) ) used to describe the dynamics of the tripod system. Thecontrol pulses Ω ˜1e ( t ) and Ω ae ( t ) perform a double STIRAPprotocol which cyclically evolves the zero-energy dark statesin the Λ system. more concretely, we follow Ref. [16], and consider controlpulses of the formΩ ( t ) = Ω cos( α ) sin[ θ ( t )] , (8a)Ω ( t ) = Ω sin( α ) sin[ θ ( t )] e iβ , (8b)Ω ae ( t ) = Ω cos[ θ ( t )] e iγ ( t ) . (8c)The angles α and θ control the relative magnitudes ofthe pulses, while β and γ control relative phases; we willonly require θ ( t ) and γ ( t ) to be time dependent. Theoverall amplitude Ω sets the instantaneous adiabatic gapof ˆ H ( t ), which we have chosen to keep constant:Ω ad ( t ) ≡ (cid:112) | Ω ( t ) | + | Ω ( t ) | + | Ω ae ( t ) | = Ω . (9)At every instant in time, ˆ H ( t ) has two zero-energy darkstates (orthogonal to | e (cid:105) ), and bright states at energy ± Ω / | d( t ) (cid:105) = cos[ θ ( t )] | ˜1 (cid:105) − e iγ ( t ) sin[ θ ( t )] | a (cid:105) (10)undergoes a cyclic adiabatic evolution | ˜1 (cid:105) → | a (cid:105) → | ˜1 (cid:105) .This requires an appropriate cyclic variation of the pulseparameter θ ( t ) (see Ref. [16] and Appendix A for de-tails). This cyclic evolution can result in a Berry phase.We take the gate to start at t = 0 and end at t = t g . Forthe case where the pulse parameter γ ( t ) is chosen as γ ( t ) = γ Θ (cid:18) t − t g (cid:19) , (11)with Θ( t ) being the Heaviside step function, this geomet-ric phase is simply γ [16].In the adiabatic limit ˙ θ ( t ) / Ω →
0, the gate unitary inthe qubit subspace is given by [16]ˆ U G , = exp ( − iγ /
2) exp (cid:16) − i γ n · ˆ σ (cid:17) , (12a) n = [sin(2 α ) cos( β ) , sin(2 α ) sin( β ) , cos(2 α )] , (12b)where ˆ σ = ( | (cid:105)(cid:104) | + H . c ., − i | (cid:105)(cid:104) | + H . c ., | (cid:105)(cid:104) | − | (cid:105)(cid:104) | )is the vector of the Pauli matrices in the qubit subspace.For example, the X gate can be realized by using theangle parameters α = π/ β = 0 and γ = π . B. Accelerated tripod gates
The geometric tripod gate yields a perfect gate fidelityin the adiabatic limit where the protocol time is infinitelylonger than 1 / Ω . In many realistic systems, dissipativeeffects involving the lower tripod levels make such longevolution times infeasible. It would thus be desirable toreduce the gate time without introducing non-adiabaticerrors. This is exactly the goal of shortcuts to adiabaticity (STA) methods [11–13, 16, 17, 29].Following Ref. [16], we consider an STA protocol basedon the SATD method [17], where non-adiabatic errors aremitigated by having the system follow a dressed adiabaticeigenstate (see Appendix B). The accelerated protocol isimplemented by simply modifying the complex envelopeof the original control pulse [16, 17, 38, 39]. Specifically,the SATD protocol requires that one corrects the originalpulses viaΩ ( t ) → ˜Ω ( t ) ≡ Ω cos( α ) (cid:34) sin[ θ ( t )] + 4 cos[ θ ( t )]¨ θ ( t )Ω + 4 ˙ θ ( t ) (cid:35) , Ω ( t ) → ˜Ω ( t ) ≡ Ω sin( α ) e iβ (cid:34) sin[ θ ( t )] + 4 cos[ θ ( t )]¨ θ ( t )Ω + 4 ˙ θ ( t ) (cid:35) , Ω ae ( t ) → ˜Ω ae ( t ) ≡ Ω e iγ ( t ) (cid:34) cos[ θ ( t )] − θ ( t )]¨ θ ( t )Ω + 4 ˙ θ ( t ) (cid:35) , (13)where the angle γ ( t ) [Eq. (11)] remains unchanged. Itcan be shown that [16] the resulting accelerated protocolobtained using correction in Eq. (13) achieves the sameunitary ˆ U G , in the qubit subspace as in the adiabaticlimit [c.f. Eq. (12)]. In what follows, we will use a tildethroughout to denote SATD-corrected pulse parameters. C. Infinite family of perfect RWA protocols
For our ideal (RWA) tripod systems, our SATD ap-proach yields an infinite number of perfect protocols (i.e.,pulse sequences) that realize a given gate in a fixed gatetime t g . These protocols are indexed by Ω (c.f. Eq. (9)),which is the scale of amplitudes of the uncorrected pulse(and the corresponding time-independent adiabatic gap). . . . time t/t g √ (cid:101) Ω e ( t ) / Ω (a) Ω (cid:28) /t g Ω ∼ /t g Ω (cid:29) /t g . . . time t/t g − (cid:101) Ω a e ( t ) / Ω (b) FIG. 3. Time profiles of the envelopes of the three driv-ing tones used to realize X gates in an ideal tripod system:(a) ˜Ω ( t ) = ˜Ω ( t ), (b) ˜Ω ae ( t ). Pulses are calculated usingEq. (13) with θ ( t ) given in Eq. (A5) of Appendix A and otherangle parameters given by α = π/ β = 0, and γ = π ..Shown here are pulses for a fixed gate time t g but with differ-ent uncorrected gap frequencies: Ω / π = 4 /t g (red curve),Ω / π = 2 /t g (green curve), and Ω / π = 0 . /t g (blue curve).The SATD correction to the adiabatic pulse becomes largeras Ω decreases. For every choice of Ω , there is a corresponding SATDprotocol [given by Eq. (13)] that yields a pulse sequencewith a perfect gate fidelity. At a heuristic level, forΩ (cid:29) /t g the uncorrected protocol is already almost inthe adiabatic limit, meaning that the additional SATDmodification of pulses will be minimal. In contrast, forΩ < /t g , the SATD correction to the original pulseshape will be extremely large (in order to cancel non-adiabatic errors). Figure 3 shows the time profiles ofpulse envelopes used to realize perfect X gates in an idealtripod system, for different choices of Ω t g . We note thatbesides the degeneracy in choosing Ω , there is also adegeneracy resulting from different choices of the pulse-shape function P ( t/t g ) [Eq. (A6)] that determines θ ( t )[Eq. (A5)]. IV. ENHANCING STA PROTOCOLS TOMITIGATE NON-RESONANT ERRORS
We now return to the central question of this work: canSTA approaches still be effective in settings where non-resonant, non-RWA processes also degrade fidelity? Inthis section, we present a general strategy for improvingSTA protocols to partially mitigate non-RWA errors. Forconcreteness, we will do this in the specific context of theaccelerated tripod gate introduced above. To achieve agate in a fixed time t g , our strategy has two basic steps:1. We first use the degeneracy of perfect STA proto-cols that exists in the RWA limit (c.f. Sec. III C) topick a protocol that minimizes the “size” (appro-priately defined) of our control fields.2. Next, we modify the SATD pulse shape by chirpingthe control fields to offset frequency-shifts arisingfrom non-RWA terms. The form of the required (a) (b) ∆ , − ∆ (1)0f , − FIG. 4. Spurious non-RWA processes giving rise to coherenterrors. (a) Crosstalk where the driving tone Ω ( t ) drivesthe | (cid:105) ↔ | e (cid:105) transition. The frequency of the driving toneΩ ( t ) is detuned by ∆ (1)0e , − from the | (cid:105) ↔ | e (cid:105) transition. (b)Coupling to spurious level | f (cid:105) where the driving tone Ω ( t )drives the | (cid:105) ↔ | f (cid:105) transition. The frequency of the drivingtone Ω ( t ) is detuned by ∆ (1)0f , − from the | (cid:105) ↔ | f (cid:105) transition. chirp can be found analytically using a perturbativeapproach.As we will see, this two-pronged, fully analytic approachresults in a modified set of pulses that yield exceptionalgate performance even when non-RWA effects are in-cluded. We now discuss each step of our general methodin more detail, focusing on the specific case of our accel-erated tripod gate. A. Step 1: Power minimization
The non-resonant, non-RWA processes described byˆ H err ( t ) (c.f. Eq. (5)) yield new unwanted coherent dy-namics that will degrade the performance of our gate;examples processes are sketched in Fig. 4. One effect ofthese terms is to generate effective time-dependent en-ergy shifts of the four levels involved in our tripod gate.We define ∆ ( j ) kl, ± = ε l − ε k ± ω j e as the detuning associ-ated with the transition between the energy level | k (cid:105) andthe level | l (cid:105) associated with the drive tone ω j e . Recallthat in our tripod gate, there is a drive tone for eachground-state level (c.f. Fig. 2), hence j = 0 , , a. Usinga Magnus-based approach (see Ref. [40]), one can derivethe leading-order, time-dependent energy shift δε k ( t ) ofenergy level | k (cid:105) due to ˆ H err ( t ). This energy shift has theusual form expected from the second-order perturbationtheory, i.e., δε k ( t ) = (cid:88) j =0 , , a σ = ± (cid:88) l | ∆ ( j ) kl,σ (cid:54) =0 | ˜ V j e ( t ) n kl | ( j ) kl,σ . (14)The sum here is over all non-resonant processes that in-volve the state | k (cid:105) . We will be interested in energy shiftsof the four tripod levels, i.e. k = 0 , , a , e. Note that theintermediate states l in Eq. (14) includes both the fourtripod levels (i.e., “crosstalk” process, Fig. 4(a)), or could involve non-tripod states (i.e., couplings to “leakage” lev-els, Fig. 4(b)). Formally, Eq. (14) is valid in the perturba-tive limit where | ˜ V j e ( t ) n kl | (cid:28) | ∆ ( j ) kl, ± | and the quasistaticlimit | (cid:82) t g ∂ t (cid:48) ˜ V j e ( t (cid:48) ) n kl e − i ∆ ( j ) kl, ± t (cid:48) dt (cid:48) | (cid:28) | ∆ ( j ) kl, ± | .The simplest way to mitigate errors associated withthe above non-RWA generated energy shifts is to mini-mize their size by minimizing the SATD-corrected pulseamplitudes ˜ V j e ( t ). We would like to find a simple metricto characterize the size of these amplitudes in a mean-ingful manner. We see that at each instant in time, therelevant quantity is the square of these amplitudes (asthe energy shifts are a second-order effect). This moti-vates characterizing the “size” of our control pulses bythe root-mean-square (RMS) voltage of the control field,i.e., ˜ V RMS ≡ (cid:115) t g (cid:90) t g (cid:104) ˜ V ( t ) (cid:105) dt. (15)Here, ˜ V ( t ) is the total real-valued control pulse function(including the SATD correction), c.f. Eq. (2).Using the specific form of the SATD pulses in Eqs. (13),we have˜ V RMS (cid:39) (cid:118)(cid:117)(cid:117)(cid:116) t g (cid:90) t g dt (cid:88) j =0 , , a (cid:12)(cid:12)(cid:12) ˜ V j e ( t ) (cid:12)(cid:12)(cid:12) = (cid:115) cos α | n | + sin α | n | + 1 | n ae | ˜Ω RMS ( t g )2 , (16)where n , n , and n ae are the tripod matrix elements,and˜Ω RMS ( t g ) ≡ (cid:115) t g (cid:90) t g dt (cid:104) | ˜Ω ( t ) | + | ˜Ω ( t ) | + | ˜Ω ae ( t ) | (cid:105) = Ω (cid:118)(cid:117)(cid:117)(cid:117)(cid:116) t g (cid:90) t g dt (cid:32) ¨ θ ( t )˙ θ ( t ) + Ω / (cid:33) . (17)In the first line of Eq. (16), we have used the fact thatthe terms involving differences of tone frequencies almostexactly average to 0. Further, in the second line, wehave use the fact that our protocol is symmetric about t = t g /
2, i.e., θ ( t ) = π/ − θ ( t − t g /
2) for t g / < t ≤ t g [Eq. (A5) in Appendix A].From Eq. (16), we can see that ˜ V RMS is related to amore fundamental metric ˜Ω
RMS / H ( t ) for a SATD-corrected pulse sequence.Note that this metric is solely a property of the ideal tri-pod Hamiltonian ˆ H ( t ) and our SATD pulse sequence.The SATD correction makes the adiabatic gap time de-pendent, and necessarily increases ˜Ω RMS ( t g ) above Ω .We finally can make use of the property discussed inSec. III C: in the RWA-limit, there are an infinite num-ber of SATD protocols that yield a perfect gate fidelity SATDDouble swap / Hybrid0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.02.02.53.03.54.0 ˜ Ω R M S t g / π Ω t g / π FIG. 5. Plot of ˜Ω
RMS t g / π vs Ω t g / π for SATD (orangecurve) and double-swap/hybrid protocols (red line). For theSATD protocol, ˜Ω RMS = Ω in the adiabatic limit (Ω t g →∞ ). Away from the adiabatic limit, the SATD correctionincreases ˜Ω RMS above Ω . The minimum of the SATD plotis ˜Ω RMS t g / π = 1 .
92 occuring at Ω t g / π = 1 . RMS t g / π = 2 is the value of ˜Ω RMS for the double-swapprotocol and hybrid scheme. for a given gate time t g . Out of this set, we will choosethe protocol that minimizes ˜Ω RMS (and hence will ap-proximately minimize the phase errors arising from thenon-RWA energy shifts δε k ( t ) in Eq. (14)). As discussed,the different SATD protocols are indexed by Ω , the adi-abatic gap associated with the uncorrected pulses. Wethus seek to identify the value of Ω t g that minimizesthe SATD energy cost ˜Ω RMS . The behavior of this quan-tity (obtained numerically for the specific smooth pulseshape function P ( t ) given in Eq. (A6)) is shown in Fig. 5.We find that ˜Ω RMS / π has a minimum value (cid:39) . /t g ,occurring for Ω / π = 1 . /t g . We have confirmed thatattempting to further optimize by using more complexpulse-shape functions P ( t ) does not yield appreciable im-provements. We have thus completed step one of our two-step approach to enhancing accelerated gates to minimizenon-RWA errors: use the degeneracy of STA protocols topick a minimum-energy pulse.Before proceeding, we pause to note something some-what remarkable: by optimizing the parameter Ω , theSATD pulses are able to achieve an energy-time tradeoff(as quantified by the product ˜Ω RMS t g ) that is better thanmore obviously fast, non-adiabatic population-transferprotocols (see red curve in Fig. 5). These are the so-calleddouble-swap protocols (see e.g., Refs. [41, 42]) (whereone sequentially moves population through the three lev-els), and the “hybrid” protocol [42] (a non-adiabatic holo-nomic protocol). This point and connections to a formalquantum speed limit valid for time-dependent Hamilto-nians [43] are discussed in Appendix C. B. Step 2: Modifying accelerated protocol pulsesto cancel non-RWA phase errors
The next step of our method is to go beyond simplyminimizing the non-RWA energy shifts δε k ( t ) in Eq. (14),and actually cancel them by slightly modifying our SATDpulses. In keeping with our general philosophy, we willderive an analytic prescription for how to do this (as op-posed to resorting to a brute-force numerical optimiza-tion).The methodology here is conceptually simple: to offsetthe unwanted, time-dependent non-RWA energy shifts,we introduce a time-dependent variation of the centraltones in our pulse (i.e., a generalized chirp). Concretely,this means that we introduce a time-dependent shift ineach of the three center frequencies ω j e ( j = 0 , , a) thatappear in our pulse: ω j e → ˜ ω j e ( t ) = ω j e − δε j ( t ) + δε e ( t ) , (18)where δε j is given by Eq. (14). These frequency shiftsensure that at each instant in time, each tone is resonant(to leading order) with the appropriate transition it isintended to drive.The net result of our approach is thus a two-step cor-rection to the original pulse in Eq. (8). For a given desiredgate time t g , we first pick an optimal value of Ω as perSec. IV A and add the SATD correction to the pulses asper Eq. (13). Subsequently, we chirp each of the threecenter frequencies as per Eq. (18). We can write theoverall modification of each control tone in Eq. (2) as V j e ( t ) exp ( iω j e t ) → ˜ V j e ( t ) exp (cid:18) i (cid:90) t ˜ ω j e ( t (cid:48) ) dt (cid:48) (cid:19) , (19)where ˜ V j e ( t ) = ˜Ω j e ( t ) /n j e are the SATD-corrected pulseenvelopes, and ˜ ω j e ( t ) are the chirp-corrected central drivefrequencies. C. Leakage errors and connections to DRAG
Our discussion so far has only focused on errors aris-ing from energy shifts generated by the non-RWA termsin ˆ H err ( t ). There is another generic kind of error toconsider: the non-RWA terms can drive transitions outof the tripod subspace, leading to a final populationof non-tripod levels. This kind of error is commonlyreferred to as leakage, and has been discussed exten-sively in many other settings (e.g., in discussing gate er-rors in weakly-anharmonic transmon-style superconduct-ing qubits [40, 44, 45]).The general approach we have taken to mitigating non-RWA errors partially minimizes leakage by minimizingthe size of the control pulses (c.f. Sec. IV A). However,we do not make any additional modifications of our pulsesto further reduce leakage errors. This is in contrast toour treatment of phase errors (which we further miti-gate through frequency chirping). It is also in contrastto the well-known DRAG technique [44, 45] for dealingwith leakage errors in superconducting circuits driven bya single control tone.There are two key rationales for our apparent neglectof pure leakage errors. The first is purely pragmatic: ingeneral there are many equally important leakage levels,and there is no simple way to modify our pulses (usingthe Magnus strategy of Ref. [40]) to simultaneously cor-rect all of these error channels. This is stark contrastto the usual DRAG problem in superconducting circuits,where there is just a single relevant leakage level (i.e., thesecond excited state of the qubit). The second rationaleis that in the perturbative limit (where non-RWA er-rors are small), leakage errors are much smaller than thephase errors associated with energy shifts. As shown inRefs. [39, 46] in the long-gate time limit ( t g (cid:29) / ∆ ( j ) kl,σ ),phase errors scales as 1 / (∆ ( j ) kl, ± t g ) , while leakage errorsare much weaker, scaling as 1 / (∆ ( j ) kl, ± t g ) .As we will see in the next section (where we imple-ment our ideas in a realistic multi-level superconductingfluxonium circuit), our approach yields extremely goodresults despite the lack of any specific leakage correction. V. FLUXONIUM QUBITA. Basic setup
The tripod gate we have analyzed is ideally suited tosystems where it is difficult to directly drive transitionsbetween the qubit levels | (cid:105) and | (cid:105) . This is often the casein qubits that have long T relaxation times. A paradig-matic example that has received considerable recent at-tention is a fluxonium-style superconducting qubit [24–27]. A fluxonium circuit consists of a single Josephsonjunction (energy E J ) in parallel with both a capacitor(charging energy E C ) and a superinductor (inductive en-ergy energy E L , typically implemented using a chain ofJosephson junctions) [Fig. 6(a)]. Experiments demon-strate that these qubits can possess exceptionally long re-laxation times (on the order of milliseconds) [25–27, 47].Moreover, they can also be made first-order insensitive tothe dephasing from 1 /f charge noise [48]. These prop-erties make fluxonium an attractive quantum comput-ing platform. A complication however is that the rela-tive isolation of qubit levels (which yields long T times)also makes conventional approaches to gates challenging.This makes our accelerated tripod gate especially attrac-tive.The fluxonium Hamiltonian isˆ H f = 4 E C ˆ n − E J cos( ˆ ϕ − π Φ ext / Φ ) + 12 E L ˆ ϕ , (20)where ˆ n and ˆ ϕ are (respectively) the charge and phase op-erators. They obey the commutation relation [ˆ n, ˆ ϕ ] = i .Φ ext is the external magnetic flux biasing the loop formedby the Josephson junction and its shunting inductance (a) (b) E J E L E C Φ ext kl ω kl / π (GHz) | n kl |
01 0.81 0.020e 8.42 0.271e 9.23 0.46ae 7.58 0.16
FIG. 6. (a) Circuit diagram of a fluxonium. (b) The qubit andtripod transition frequencies ( ω jk ) as well as the magnitudeof the charge matrix elements ( n jk ) of the fluxonium whoseenergy spectrum is given in Fig. 1. The computational statesin the tripod system are labeled by | (cid:105) , | (cid:105) , | a (cid:105) , and | e (cid:105) . Thefluxonium parameters used to get the values of ω jk and n jk inpanel (b) are E L /h = 0 .
063 GHz, E J /h = 9 .
19 GHz, E C /h =2 GHz and Φ ext = 0 . . and Φ = h/ e is the flux quantum. Eq. (20) shows thatthe effective phase potential consists of a cosine poten-tial superimposed on a parabolic background [see Fig. 1].The highly tunable, anharmonic nature of the fluxoniumallows us to engineer a variety of different candidate four-level tripod systems. B. Optimal parameters for a tripod gate
A first question is to identify parameters yielding an“optimal” tripod configuration, meaning that we bothhave a long T time as well as an accelerated SATD gatewith small (non-RWA) coherent errors. This leads to thefollowing selection criteria:1. The ground states should be well isolated from eachother (i.e., small charge matrix element connectingthem) and be nondegenerate. Strong isolation en-sures a long T time.2. The charge matrix elements coupling the excitedstate to the ground states of the tripod system mustbe large and have the same order of magnitude.The latter helps minimize coherent errors arisingfrom non-RWA processes.Since the above requirements cannot be perfectly satis-fied simultaneously (see Appendix D), we choose pa-rameters that strike an optimal balance between them.To this end, we performed a numerical search throughparameter space to identify optimal regimes. The re-sult is the following near-ideal parameter set for realizinghigh-fidelity SATD tripod gates in a T protected regime: E L /h = 0 .
063 GHz, E J /h = 9 .
19 GHz, E C /h = 2 GHzand Φ ext = 0 . (see Appendix D for parameter justi-fication). We note that the small inductive energy hereputs our device in same regime as the “Blochnium” cir-cuit recently realized in experiment [49].For the above parameter set, the qubit and tripodtransition frequencies together with their correspondingcharge matrix elements are shown in Fig. 6(b). The corre-sponding energy spectrum and wavefunctions are plottedin Fig. 1. We have labeled the energy levels used for thetripod system by | (cid:105) , | (cid:105) , | a (cid:105) , and | e (cid:105) . The charge matrixelement connecting qubit states | (cid:105) and | (cid:105) is extremelysmall as desired: | n | ≡ |(cid:104) | ˆ n | (cid:105)| = 0 .
02. In contrast,the charge matrix elements for the desired tripod tran-sitions are much larger and comparable in magnitude toone another: | n | = 0 . | n | = 0 .
46 and | n ae | = 0 . C. SATD protocols for tripod gates in fluxonium
To realize our accelerated tripod gate, we drive thefluxonium circuit with a microwave pulse (described bya voltage V ( t )) that couples to the charge operator ˆ n .In the eigenbasis of the bare fluxonium Hamiltonian ˆ H f [Eq. (20)], we can write the Hamiltonian of the drivenfluxonium exactly in the general form given in Eq. (1).To realize the accelerated tripod gate, the driving voltage V ( t ) [Eq. (2)] consists of three driving tones V j e ( t ) =Ω j e ( t ) /n j e for j = 0 , , a [see Eq. (7)]. Here, Ω j e ( t ) is thecomplex coupling given in Eq. (8) for the uncorrectedpulse and Eq. (13) for the SATD corrected pulse. time (ns) − e Ω j e ( t ) / π ( M H z ) e Ω ( t ) / π = e Ω ( t ) / π e Ω ae ( t ) / π time (ns) e ω j e ( t ) / π ( G H z ) e ω ( t ) / π e ω ( t ) / π e ω ae ( t ) / π ( a )( b ) FIG. 7. Time profiles of (a) the envelopes and (b) chirpedfrequencies of the three driving tones used to realize SATD Xgates in a realistic fluxonium. The pulse sequences in panel(a) are given in Eq. (E1). They are obtained by sandwichingthe pulses in Eq. (13) by a ramp time t ramp at the beginningand end of the protocol during which the ˜Ω ae ( t ) pulse (greencurve in panel (a)) is turned on and off, respectively, using asmooth polynomial function given in Eq. (A6). The param-eters used are t g = 100 ns, turn-on/off time t ramp = 1 ns,Ω / π = 1 . /t g = 11 .
35 MHz. The parameter set for thefluxonium is the same as that used in Fig. 6.
The envelopes of the driving tones ˜Ω j e ( t ) of theSATD tripod gate pulse (for optimal Ω t g ) are shown in Fig 7(a). We have slightly modified the pulses derived inSec. III B so that V ( t ) goes smoothly to zero at the startand end of the protocol (as would be in the case in exper-iment). We do this by sandwiching the ideal pulses witha short ramp up (down) of duration t ramp = 0 . t g atthe beginning (end) of the protocol. During this ramp,the ˜Ω ae ( t ) tone is smoothly turned on and off, respec-tively [see Appendix E]. We specifically use Eq. (A6) fora smooth ramp function that turns the pulse on and off.Note that including these ramps does not appreciablychange our results. VI. GATE PERFORMANCE: COMPARINGDIFFERENT ERROR CHANNELS
To quantify the performance of our accelerated tripodgate, we calculate the state-averaged fidelity of the gate.This is given by [50]¯ F = 16 (cid:88) m = ± x, ± y, ± z Tr (cid:104) ˆ U q ˆ ρ m ˆ U † q ˆ ρ m ( t g ) (cid:105) , (21)where ˆ ρ m is an axial pure state on the qubit’s Blochsphere with m ∈ {± x, ± y, ± z } (e.g. ˆ ρ x = 1 / | (cid:105) + | (cid:105) )( (cid:104) | + (cid:104) | )). ˆ ρ m ( t g ) is the lab-frame density matrixof the system at the end of the protocol ( t = t g ) for theinitial state ˆ ρ m . Here, ˆ U q is a product of the ideal tar-get unitary gate operation in the qubit subspace ˆ U G , [c.f. Eq. (12)] and an innocuous phase factor correspond-ing to dynamical phases in the lab-frame:ˆ U q = ˆ U diag , ( t g ) ˆ U G , , (22)with ˆ U diag , ( t g ) = (cid:80) k =0 , e − i (cid:82) tg ε k ( t ) dt | k (cid:105)(cid:104) k | . Here, ε k ( t ) = ε k for unchirped protocols and ε k ( t ) = ε k + δε k ( t )for chirped protocols. In what follows, we use this stan-dard metric to characterize a target X qubit gate in thepresence of both coherent (non-RWA) errors and dissipa-tion. A. Effects of coherent errors only
Consider first the case where dissipation is neglected,and the only source of gate errors are the non-RWAterms in Eq. (5). To calculate gate performance in thislimit, we numerically evolve initial states as per the lab-frame Hamiltonian [Eq. (1)] using the Python packageQuTiP [51, 52]. We performed the simulation by includ-ing the 18 lowest energy levels of the fluxonium and allcharge matrix elements in this space. We have checkedthat including more energy levels in the simulations doesnot change the results.In Fig. 8, we plot the gate errors ¯ ε = 1 − ¯ F for a targettripod gate ˆ U G , = − ˆ σ x, (an X-gate) obtained for dif-ferent choices of pulses. For all curves, the uncorrectedgap frequency at each protocol time t g is picked to be theoptimal value Ω / π = 1 . /t g (see Sec. IV A). If weuse the uncorrected adiabatic pulses [c.f. Eq. (8), greencurve], errors arise both from non-adiabatic transitionsas well as non-RWA processes. If instead, we use theSATD-corrected pulses [purple curve, c.f. Eq. (13)], non-adiabatic errors are completely canceled, leaving onlynon-RWA errors. Finally, if we also frequency chirp theSATD pulses as per Eq. (18) (red curve), we further re-duce gate errors by reducing the leading non-RWA errors.This yields a dramatic improvement over the unchirpedSATD protocol at long gate times.As discussed, our corrections do not specifically can-cel pure leakage errors. To characterize these, we cal-culate the state-averaged population outside the tripodsubspace (leakage population) at the end of the protocol.This is given by1 − ¯ P tripod = 1 − (cid:88) m = ± x, ± y, ± z Tr (cid:104) ˆ P tripod ˆ ρ m ( t g ) (cid:105) , (23)where ¯ P tripod and ˆ P tripod are respectively the state-averaged population and projector in the tripod sub-space. The final state-averaged leakage population isplotted as a gray dashed curve in Fig. 8. As discussed,leakage makes a minimal contribution to error at modestto long gate times. Gate time t g (ns) − − − − − − − − X gate, coherent errors only
Uncorrected gate errorSATD gate errorSATD chirp gate errorSATD chirp leakage pop
FIG. 8. State-averaged gate error ¯ ε = 1 − ¯ F (solid curves)vs gate time t g for different realizations of a tripod X gatein a fluxonium qubit, in the absence of dissipation. We keepthe lowest 18 levels of the circuit, and include all non-RWAerror channels. Curves correspond to uncorrected adiabaticpulses (green), SATD pulses without frequency chirping (pur-ple) and SATD pulses with frequency chirping (solid red). Forall pulses, we use the optimal value of the uncorrected gap fre-quency Ω / π = 1 . /t g that minimizes the RMS voltage.Also shown is the leakage population outside the tripod sub-space for the chirped SATD protocol (dashed gray). B. Effects of /f dephasing noise only
1. Effective modelling of non-Markovian noise
We now turn to modelling additional gate errors aris-ing from dissipation. Given our operating point (isolatedqubit states, but not at the flux sweet spot), 1 /f fluxnoise will be the dominant form of dissipation. As iscommon [53–57], we will model this non-Markovian noiseusing an approximate Markovian description that quali-tatively captures the relevant dephasing timescales cor-rectly. Letting ˆ ρ ( t ) denote the fluxonium reduced densitymatrix, we model our system by the Lindblad-form mas-ter equation ∂ t ˆ ρ ( t ) = − i [ ˆ H ( t ) , ˆ ρ ( t )] + (cid:18) ˆ Z ˆ ρ ˆ Z − { ˆ Z , ˆ ρ } (cid:19) . (24)ˆ H ( t ) is the driven fluxonium Hamiltonian [Eq. (1)], andthe Hermitian operator ˆ Z has the general form:ˆ Z = (cid:88) k sgn (cid:18) ∂ε k ∂ Φ ext (cid:19) (cid:112) k | k (cid:105)(cid:104) k | . (25)Heuristically, this describes the fact that each fluxoniumenergy level ε k depends on the bias flux, and hence fluxnoise causes each energy to fluctuate. We have writtenthe coupling constant associated with each level | k (cid:105) interms of an overall sign (which captures whether ε k in-creases or decreases with increasing flux), and a magni-tude Γ k .To fix the couplings Γ k , we use the fact that forclassical, Gaussian 1 /f noise, free induction decay ofa given coherence ρ kl ≡ (cid:104) k | ˆ ρ | l (cid:105) has a decay envelopeexp (cid:2) − ( t/T ϕ,kl ) (cid:3) (up to logarithmic corrections). Astandard calculation (see e.g., Ref. [55]) yields:1 /T ϕ,kl = A Φ ext | ∂ Φ ext ( ε k − ε l ) | (cid:112) | ln D | . (26)Properties of the flux-noise spectral density only enterthrough A Φ ext (the standardly-defined flux-noise ampli-tude) and D (the product of a measurement time andthe low-frequency cutoff of the noise, see Refs. [28, 55]).We will construct our approximate Markovian masterequation by picking the couplings Γ k to ensure that forfree induction decay over a time t g , the final decay ofa large set of coherences is captured correctly. In par-ticular, we take | (cid:105) as a reference level (i.e., the lowestenergy level of the fluxonium), and insist that the net de-cay of any coherence ρ k ( k (cid:54) = 1) after an evolution time t g is the same for our Markovian dynamics as it would befollowing the non-Markovian, Gaussian-lineshape decaydescribed by Eq. (26). This leads to the choice Γ = 0,and for k (cid:54) = 1: Γ k = Γ k ( t g ) = t g / ( T ϕ,k ) . (27)We stress that Γ k (and hence our master equation) de-pends only on the choice of total evolution time t g ; duringthe evolution the dephasing superoperator is constant.0The above choice guarantees that at the end of evo-lution for a time t g , the overall free induction decay ofany coherence involving the qubit level | (cid:105) (the lowestenergy state of the circuit) is captured correctly. Asdiscussed in Appendix F, there is no way to make achoice for the Γ k that captures the decay of all coher-ences correctly. Nonetheless, as shown in the appendix,our approach if anything overestimates the dominant de-phasing within the tripod subspace (see Table I of Ap-pendix F). Note also that our approach overestimates de-phasing compared to alternate approximations that usean explicitly time-dependent Γ k [56, 57].In what follows we take A Φ ext = 3 µ Φ (which is typ-ical for state-of-the-art experiments [26, 27, 58]), andalso choose (following Refs. [28, 55, 59]) the constant D = (2 π × × µs . With these choices and forthe circuit parameters used here, we find dephasing times T ϕ,kl that are of the order ∼ − µ s [Table I of Ap-pendix F]. The dephasing of the qubit levels is T ϕ, = 7 µ s. Note that we have confirmed that our results arelargely unchanged if one adds realistic T decay pro-cesses (e.g., dielectric loss with dielectric quality factor Q diel (cid:38) ). This is not surprising. The tripod gate in-volves the dynamics of eigenstates of our driven Hamilto-nian; as they are superpositions of fluxonium eigenstates | k (cid:105) , dephasing processes are able to cause transitions be-tween them.
2. RWA gate performance in the presence of dephasingnoise
We numerically evolve the master equation Eq. (24)(with the above form for the dephasing superoperator),and use the results to calculate the state-averaged gateerror ¯ ε = 1 − ¯ F of a tripod X gate (c.f. Eq. (21)). We firstconsider the case where all non-RWA terms in the coher-ent Hamiltonian are neglected, so that errors are only theresult of dissipation or non-adiabaticity. The results asa function of gate time t g are shown in Fig. 9, where theperformance of the uncorrected and accelerated (SATD)pulses are compared. The simulations of the gate dynam-ics for all gate times t g are done for a fixed Ω / π = 100MHz.The behavior here is generic and as expected. In theadiabatic regime where t g (cid:29) / Ω , the uncorrected andSATD gates have almost identical performance. In thisregime the error ¯ ε is dominated by dephasing, and growsquadratically with t g (reflecting the quadratic loss of co-herence expected from 1 /f noise at short times). Thisquadratic-in- t g error scaling continues down to small val-ues of t g for the SATD curve, as in this case, dephasing isthe only error mechanism. In contrast, the uncorrectedcurve has much larger errors at short time, correspondingto non-adiabatic errors. There are special sharply definedvalues of t g where these non-adiabatic errors construc-tively cancel; as discussed in Ref. [16], these are difficultto exploit experimentally as they require extreme fine Gate time t g (ns) − − − − − − − G a t ee rr o r ¯ ε X gate, ideal tripod (RWA) with /f flux noise UncorrectedSATD
FIG. 9. State-averaged gate error ¯ ε = 1 − ¯ F as a function ofgate time t g for ideal (4 level, RWA) tripod X gates. We com-pare uncorrected adiabatic pulses (orange) against the accel-erated SATD pulses (red), in the presence of 1 /f flux noise.The scale of the uncorrected pulses is set by Ω / π = 100MHz, and the 1 /f flux noise has a strength parameterized by A Φ ext = 3 µ Φ (yielding a qubit dephasing time T ϕ, = 7 µ s).Fluxonium parameters match those given in Fig. 6. tuning. VII. FULL GATE PERFORMANCE ANDCOMPARISON AGAINST BRUTE-FORCEDIRECT DRIVINGA. Direct driving gate and power scaling
Having investigated the impact of different error chan-nels (non-adiabatic errors, non-RWA errors, dephasing),we are now ready to study the accelerated tripod gatewith all error channels present. To properly understandthe advantages of our accelerated tripod gate, it is in-structive to compare performance against the simplestpossible gate. This is the “direct driving” (DD) gate,where one resonantly drives the qubit transition between | (cid:105) and | (cid:105) . As we are working with a T protected qubit,the magnitude of the matrix element for this transition | n | is small; however, for a large enough power onecould in principle still achieve a given gate. To compareour tripod approach against this approach, we will wantto compare not only the gate error ¯ ε , but also the power required to achieve the gate. As we will see in whatfollows, in many experimental systems additional con-straints limit the magnitude of pulses that can be used.This will provide a strong advantage in many regimes forthe tripod gate.We begin by writing a simple pulse shape that can beused to realize a DD gate [39]:˜ V DD ( t ) = χt g | n | (cid:20) − cos (cid:18) πtt g (cid:19)(cid:21) cos (cid:20)(cid:90) t dt (cid:48) ˜ ω ( t (cid:48) ) (cid:21) , (28)1where χ = π for an X gate. Similar to our acceleratedadiabatic gate, we will chirp to partially correct for non-RWA errors, and hence ˜ ω ( t ) = ω + δε ( t ) − δε ( t )[c.f. Eq. (18)].Using the definition in Eq. (30), we can calculate theRMS time-averaged voltage of the DD gate. We canalso use our previous result [Eq. (16)] for the RMS time-averaged voltage of the SATD gate (using the optimalvalue of Ω t g discussed in Sec. IV A). These two RMSvoltages are given by˜ V RMS , DD = √ χ | n | t g, DD , (29a)˜ V RMS , SATD = 1 . πt g, SATD (cid:115) cos α | n | + sin α | n | + 1 | n ae | . (29b)In both cases, the RMS voltage scales inversely with thegate time t g , but note the crucial dependence on matrixelements and gate type.It follows from Eq. (29a) that an X gate is the mostenergy-consuming (and hence problematic) gate if oneuses the DD approach. We will thus focus on this gate inwhat follows, and compare against our accelerated adi-abatic approach. Substituting in the matrix elementsfor the fluxonium parameters used throughout this paper[see Fig. 6(b)], we get the ˜ V RMS for the X gate ( χ = π for DD and α = π/ V RMS , DD = 136 t g, direct , (30a)˜ V RMS , SATD = 42 . t g, SATD . (30b)The equations already describe a crucial advantage ofthe tripod approach over direct driving: for a fixed gatetime, the SATD tripod approach requires drive ampli-tudes that are 3.2 times smaller, corresponding to a fac-tor of ≥ H c = (cid:88) kl gn kl | k (cid:105)(cid:104) l | (ˆ a † + ˆ a ) . (31)Here g is the cavity-fluxonium coupling, n kl is a chargematrix element, and ˆ a is the cavity photon annihilationoperator. A standard constraint in such setups is thatthe time-averaged intracavity photon number ¯ n cav shouldnot exceed some small value [58, 60] to avoid additionaldissipative mechanisms; here, we will require ¯ n cav ≤ . V = 2¯ n cav g ≤ (0 . g . (32) Gate time t g (ns) − − − − − − − − G a t ee rr o r ¯ ε X gate, coherent errors only
SATDdirect
55 60
Gate time t g (ns) . . . G a t ee rr o r ε ¯ D FIG. 10. State-averaged gate error ¯ ε = 1 − ¯ F as a function ofgate time t g , for SATD tripod (red curve) and direct-driving(blue curve) realizations of a fluxonium X gate. Dissipationis not included here, but non-RWA error channels are. Inset:zoom-in of the SATD gate error. The gates are calculatedusing the 18 lowest energy levels of the circuit. For the SATDgate, we use the optimal uncorrected gap frequency Ω / π =1 . /t g that minimizes the RMS voltage at each gate time t g . The fluxonium parameters are as in Fig. 6. In what follows, we will compare the SATD tripod gateagainst the DD gate for fixed values of g (and hencefixed maximum possible ˜ V RMS ). We will see that thisphysically-motivated power constraint gives the acceler-ated adiabatic gate an important advantange. Furtherdetails about driving via cavity (including the drivingpulse applied to the cavity and constraints that allowto ignore cavity-induced dissipation) are given in Ap-pendix G.
B. Comparison of gate performance, coherenterrors only
We first compare the accelerated tripod gate to thedirect driving (DD) gate in the absence of dissipation,but including all non-RWA terms. Figure 10 shows thestate-averaged gate errors ¯ ε for an X gate as a functionof gate time t g for our chosen fluxonium parameters. Tomitigate non-RWA errors, the SATD protocol is imple-mented using the optimal value of Ω (see Sec. IV A) andis frequency chirped (see Sec. IV B); the DD gate is alsochirped.We see that for both approaches, errors increase as t g isreduced. This simply reflects the higher drive amplitudesneeded at shorter times (which in turn increase non-RWAerrors). For all gate times, the coherent errors are largerfor SATD versus DD. This is a result of the SATD proto-col using multiple drive tones, and being subject to morenear-resonant non-RWA error channels (including thoseinvolving higher-energy excited states). Note that inter-ference between its multiple drive tones causes the SATDerror curve to exhibit fast, low-amplitude oscillations as2 (cid:71)(cid:97)(cid:116)(cid:101) (cid:116)(cid:105)(cid:109)(cid:101)(cid:44) t g (cid:40)(cid:110)(cid:115)(cid:41) − − − − (cid:71) (cid:97) (cid:116) (cid:101)(cid:101) (cid:114)(cid:114) (cid:111) (cid:114) (cid:22) (cid:34) SATDdirect
60 100 140 180 220 260 300 (cid:71)(cid:97)(cid:116)(cid:101) (cid:116)(cid:105)(cid:109)(cid:101) t g, SATD (cid:40)(cid:110)(cid:115)(cid:41) (cid:58) (cid:58) (cid:58) (cid:71) (cid:97) (cid:116) (cid:101)(cid:101) (cid:114)(cid:114) (cid:111) (cid:114) (cid:22) (cid:34) g / π = M H z g / π = M H z (cid:58) (cid:58) (cid:58) (cid:58) (cid:58) (cid:58) (cid:58) (cid:71)(cid:97)(cid:116)(cid:101) (cid:116)(cid:105)(cid:109)(cid:101) t g, DD (cid:40)(cid:110)(cid:115)(cid:41)(b) Including power constraintX gate (with /f dephasing and non-RWA errors)(a) No constraints on power FIG. 11. Comparison between state-averaged X-gate error ¯ ε = 1 − ¯ F for SATD (red curve) and direct-driving (blue curve)protocols in our fluxonium system, including effects of 1 /f flux noise dephasing as well as non-RWA errors; 18 fluxoniumlevels are included in the simulations. Dephasing is treated as per Sec. VI B 1, with a 1 /f flux noise amplitude A Φ ext = 3 µ Φ corresponding to the qubit dephasing time T ϕ, = 7 µ s. (a) Log-log plot of gate error ¯ ε vs gate time t g . For each time,the DD protocol has a smaller error but uses significantly larger pulse amplitudes. (b) Linear plot of gate error ¯ ε for bothprotocols at equal power levels ˜ V RMS [c.f. Eq. (30)]. The shaded (unshaded) region corresponds to the time regime where thegate errors are dominated by non-RWA errors (dephasing). For the same value of ˜ V RMS , the gate time for the direct driving(DD) protocol is longer than the SATD protocol (hence the two distinct x axes). Vertical lines indicate the shortest gate timespossible (corresponding to the maximum allowed ˜ V RMS , see Eq. (32)) for cavity-based driving for time-averaged cavity photonnumber ¯ n cav = 0 .
05 and a fixed cavity-fluxonium coupling g (purple vertical lines). Fluxonium parameters are the same as inFig. 6. a function of t g (see inset of Fig. 10).The results of Fig. 10 may seem depressing. However,as we will argue in what follows, they are misleading. Fora given gate time, we have already seen (c.f. Sec. VII)that DD requires considerably larger pulse amplitudesthan SATD. Once we enforce power constraints asso-ciated with realistic driving via a cavity (and also in-clude dissipation), the accelerated tripod gate will havea marked advantage. C. Qubit gates with /f flux noise We next compare the DD gate and accelerated SATDtripod gate including 1 /f flux noise (modelled as perSec. VI B 1), as well as non-RWA errors; results areshown in Fig. 11(a). For both protocols, the error isnon-monotonic with gate time t g . For short times (greyshaded region) errors are dominated by non-RWA effectsand decrease with increasing t g . For longer times dephas-ing dominates, causing error to increase with increasing t g . In this latter regime, both curves increase quadrat-ically, but the SATD curve has the higher error. Thiscorresponds to a shorter decoherence timescale for tri-pod coherences (as used in SATD) versus the qubit 01coherence (see Table I in Appendix F).In Fig. 11(b) we replot these results in a way that now accounts for power constraints that arise when drivingthrough a cavity (c.f. Sec.VII). Each vertical cut corre-sponds to a fixed value of RMS voltage (c.f. Eq. (30));given that DD uses more power, fixing voltage thus re-sults in a t g for SATD that is smaller by ∼ . t g axis for DD versus SATD). We thus see that once power isconstrained (through ˜ V RMS ), the accelerated tripod gatehas a marked advantage over DD for t g (cid:38)
100 ns; this isthe regime where dephasing dominates non-RWA errors,and hence the faster speed of SATD is advantageous.We can quantify this relative advantage by the ratio ofDD and SATD errors for the same fixed ˜ V RMS : ζ ≡ ¯ ε DD ( ˜ V RMS )¯ ε SATD ( ˜ V RMS ) . (33)In the dephasing-limited regime, the gate errors for theSATD and DD gates both exhibit a quadratic scalingwith t g . For our chosen parameters and 1 /f flux noisestrength, we find ζ = ¯ ε DD ¯ ε SATD = 5 . . (34)We thus have a central conclusion of our work: ouranalytically designed accelerated tripod gate allows oneto suppress gate errors (associated with an X gate) by3more than a factor of 5 compared to more simplistic DDapproach. This conclusion holds for ˜ V RMS values smallenough that errors are dominated by dephasing. As dis-cussed in Sec.VII, the maximum value of ˜ V RMS is deter-mined by the cavity-qubit coupling g and the require-ment that the time-averaged intracavity photon number¯ n cav ≤ .
05. As shown in Fig. 11 (purple vertical lines),realistic choices of g put us squarely in this dephasing-dominated regime where we have a strong advantage.While we have focused on comparing our acceleratedtripod gate against DD, it is also worthwhile to considerits absolute performance. As shown in Fig. 11(a), we areable to achieve a fidelity of approximately 0 . t g = 100 ns (including 1 /f flux noise with A Φ ext =3 µ Φ and non-RWA error channels). This compares wellto gates designed for similar systems using state-of-the-art numerical optimal control methods. For example,Ref. [28] used numerical optimal control to design gatesin a fluxonium circuit operated in a T -protected regime.This study constrained drive power in a similar mannerto our approach, but used slightly different fluxoniumparameters and a lower level of flux noise ( A Φ ext = 1 µ Φ ,3 times smaller than in our work). Ref. [28] achieved anX gate with ¯ F ≈ .
996 of in a time t g = 60 ns. The maindifference in parameters is that Ref. [28] used a flux biasΦ ext = 0 . (versus Φ ext = 0 . in our work), andhad more isolated qubit states: | n | = 0 .
01 in their work,a factor of two smaller than in our model system.
VIII. CONCLUSIONS
In this paper we have presented a general strat-egy showing how analytic shortcuts-to-adiabaticity tech-niques can be employed even in complex multi-level sys-tems where the rotating-wave approximation is not valid.Our strategy to mitigate non-RWA errors involved firstexploiting the degeneracy of perfect STA protocols in theRWA limit, and then correcting pulses with (analytically-derived) frequency chirps. As a demonstration of ourtechnique, we theoretically analyzed the implementationof an accelerated adiabatic “tripod” gate in a realisticmultilevel superconducting circuit (a driven fluxoniumqubit). We focused on parameter regimes where thequbit levels are highly isolated, yielding T protection butmaking traditional gates more problematic. Our analysisrevealed that our techniques combined with a judiciouschoice of system parameters yield an accelerated adia-batic gate having competitive performance. Includingrealistic levels of 1 /f flux noise dephasing, we achieve agate fidelity of 0.9997 in a gate time of 100 ns. We alsoshowed that our approach can compare favorably againsta more straightforward direct-driving approach to gatesin this system. Using power constraints arising from a re-alistic setup as if the qubit is driven through a cavity, wefound that for an X gate, our approach yields errors morethan 5 times smaller than the direct driving approach.While our test system consisted of a fluxonium qubit, our discussion and methods are very general and canbe readily applied to various other architectures. This,along with the fact that the pulses are described analyt-ically, while still providing good performance, will hope-fully prove to be very useful in various quantum-control-related applications. ACKNOWLEDGMENTS
This work is supported by the Army Research Officeunder Grant Number W911NF-19-1-0328. We thank He-lin Zhang, Srivatsan Chakram, Brian Baker and JensKoch for fruitful discussions. F.S. is grateful to LongB. Nguyen for enlightening discussions and pointing outRefs. [58, 60]. We acknowledge the University of ChicagoResearch Computing Center for support of this work.
Appendix A: Basic working of the tripod gate
In this section, we give a brief overview of the basic4-level geometric tripod gate introduced in Refs. [3, 4][see Fig. 2(a)]; the discussion here follows Ref. [16]. Thetripod Hamiltonian ˆ H ( t ) [Eq. (6)] has two instantaneouszero-energy dark states which span the dark-state man-ifold and are orthogonal to | e (cid:105) . The basic tripod gateuses the geometric evolution of states in the dark statemanifold. In this manifold, there is always one (time-independent) state that is purely qubit-like: | ˜0 (cid:105) = sin( α ) | (cid:105) − exp( iβ ) cos( α ) | (cid:105) . (A1)The qubit state orthogonal to this dark state is | ˜1 (cid:105) = cos( α ) | (cid:105) + exp( iβ ) sin( α ) | (cid:105) . (A2)Expressing ˆ H ( t ) in these new qubit basis states, we haveˆ H ( t ) = 12 (cid:2) Ω ˜1e ( t ) | ˜1 (cid:105)(cid:104) e | + Ω ae ( t ) | a (cid:105)(cid:104) e | + H . c . (cid:3) , (A3)where Ω ˜1e = Ω sin[ θ ( t )] and Ω ae = Ω cos[ θ ( t )] e iγ ( t ) [Eq. (8c) of the main text]. In this new basis, the qubitstate | ˜1 (cid:105) , together with states | a (cid:105) and | e (cid:105) , form a three-level Λ system [36, 37] [see Fig. 2(b)]. As is well known,one can write a geometric phase [61, 62] onto the | ˜1 (cid:105) byperforming the “double STIRAP protocol”, where oneslowly varies control pulses to realize the cyclic adiabaticevolution | ˜1 (cid:105) → | a (cid:105) → | ˜1 (cid:105) . The resulting phase is thebasis of the tripod adiabatic single-qubit gate [3, 4]. Onecan perform an arbitrary single qubit gate in this manner,without requiring precise pulse timing, and without re-quiring direct couplings between the logical qubit states.To understand the above double-STIRAP protocol inmore detail, note that the dark state relevant to our Λsystem (and orthogonal to | ˜0 (cid:105) ) is: | d( t ) (cid:105) = cos[ θ ( t )] | ˜1 (cid:105) − e iγ ( t ) sin[ θ ( t )] | a (cid:105) . (A4)4The required cyclic adiabatic evolution is achieved byvarying the pulse parameter θ ( t ), which brings the darkstate | d( t ) (cid:105) from the state | ˜1 (cid:105) at t = 0 to | a (cid:105) at t = t g / | ˜1 (cid:105) at the final gate time t = t g . To do this,we use a symmetric form for θ ( t ), i.e., θ ( t ) = π P ( t/t g ) 0 ≤ t ≤ t g π (cid:20) − P (cid:18) tt g − (cid:19)(cid:21) t g < t ≤ t g , (A5)where P ( x ) is a function that increases monotonicallyfrom P (0) = 0 to P (1 /
2) = 1. Furthermore, to ensure asmooth turn on and off of the control fields, we choose apolynomial which gives ˙ θ (0) = ˙ θ ( t g /
2) = ˙ θ ( t g ) = ¨ θ (0) =¨ θ ( t g /
2) = ¨ θ ( t g ) = 0. In particular, we use the simplestpolynomial satisfying the above criteria which is givenby [16] P ( x ) = 6 (2 x ) −
15 (2 x ) + 10 (2 x ) . (A6)One also needs a non-trivial relative pulse phase γ ( t ) toobtain a net Berry phase. Following Ref. [16], we use thesimple form of γ ( t ) as given in Eq. (11) of the main text: γ ( t ) = γ Θ (cid:18) t − t g (cid:19) , (A7)where Θ( t ) is the Heaviside step function.In the adiabatic limit ˙ θ ( t ) / Ω →
0, one can showthat [16] the dark state | d( t ) (cid:105) accumulates a geometricphase γ at t = t g , and qubit subspace evolves inde-pendently of the auxiliary-level subspace. The net re-sult is a geometric single-qubit gate controlled by thepulse parameters α, β , and γ . It is described by theunitary in the qubit subspace ˆ U G , given in Eq. (12)of the main text. The full adiabatic limit unitary hasthe form ˆ U G = ˆ U G , ⊕ ˆ U G , ae where ˆ U G , and ˆ U G , ae arethe unitaries acting in the qubit and auxilary subspaces,respectively (see Ref. [16]). Appendix B: SATD dressing for STA protocols
We review briefly how the “dressed state” approach toconstructing STA protocols [16, 17] can be used to ac-celerate the tripod gate [16]. The general goal is to havethe system follow a “dressed” version of the original adi-abatic eigenstate which coincides with the original stateat the start and end of the protocol. This can be achievedby using a time-dependent dressing function ν ( t ) whichvanishes at t = 0 and t = t g . Following Refs. [16, 17],we introduce | d ν (cid:105) , a dressed version of the original darkstate | d (cid:105) [Eq. (10)]: | d ν ( t ) (cid:105) = exp (cid:104) − iν ( t ) ˆ J x (cid:105) | d( t ) (cid:105) . (B1)Here ˆ J x = ( | b + ( t ) (cid:105)(cid:104) d( t ) | + | b − ( t ) (cid:105)(cid:104) d( t ) | + H . c . ) / √
2, and | b ± ( t ) (cid:105) denote the bright adiabatic eigenstates of ˆ H ( t )with energies ± Ω /
2. As discussed in Ref. [16], for the phase accumulatedby this state to be purely geometric and equal to theadiabatic-limit geometric phase γ , we require ν ( t g /
2) =0. A particular dressing that satisfies this constraint isthe “superadiabatic transitionless driving” (SATD) [16,17] where the dressing angle ν is given by ν ( t ) = ν SATD ( t ) ≡ arctan (cid:34) θ ( t )Ω (cid:35) . (B2)Using the SATD dressing function, one can show that [16,17] the accelerated protocol is implemented by modifyingthe original uncorrected pulse according to Eq. (13) of themain text. Appendix C: Comparing SATD against simplenon-adiabatic protocols
In Sec. IV A, we discussed how ˜Ω
RMS , the time-averaged adiabatic gap of our Hamiltonian, was a rel-evant metric of our protocol’s energy cost. We also dis-cussed that by optimizing SATD, one could achieve anenergy cost ˜Ω
RMS / π = 1 . /t g where t g is the gate time.In this appendix, we show that this compares surprisinglyfavorably to simpler, non-adiabatic pulse protocols.First, consider the double-swap and hybrid scheme pro-tocols (see e.g., Refs. [41, 42]). To understand theseprotocols, consider the effective three-level Λ systemas shown in Fig. 2(b) [with the Hamiltonian given inEq. (A3) of Appendix A]. The double-swap protocol in-volves two sequential swap operations for each STIRAPprocess where in the first half of the protocol, the pulseΩ ˜1e ( t ) is first turned on with a constant value of Ω forhalf of the time and then turned off with a simultaneousturn on of the pulse Ω ae ( t ) with a constant value of Ω for another half of the time. The whole sequence is thenreversed for the second half of the protocol. It followsthat to generate a geometric quantum gate which cycli-cally evolves the state | ˜1 (cid:105) → | e (cid:105) → | a (cid:105) → | e (cid:105) → | ˜1 (cid:105) ,the protocol must be executed for a total gate time t g = 2 π/ (Ω / RMS = Ω = 4 π/t g .As shown in Fig. 5, this is slightly larger than the energycost of the optimized SATD protocol.Alternatively, consider the hybrid scheme. This hasboth pulses [Ω ˜1e ( t ) and Ω ae ( t )] turned on for the wholeprotocol with a constant value of Ω . For a cyclic evo-lution of the state, the protocol must then be carriedout for a total gate time of t g = 2 π/ ( √ / RMS = √ = 4 π/t g which is the same as the double-swap protocol.
1. Proof for the lower bound of ˜Ω RMS
Finally, we establish a rigorous lower bound on ˜Ω
RMS using the quantum speed limit of Ref. [43], which gen-eralized previous works [63, 64]. The lower bound on5˜Ω
RMS that we are going to derive below holds for anygeneric dressing function ν ( t ) [Eq. (B1)] including theSATD dressing [Eq. (B2)]. We begin by applying thebound of Ref. [43] to the first half of our gate protocol.Letting ˆ ρ (ˆ ρ t g / ) denote the initial system state (stateafter evolution for a time t g / L QF (ˆ ρ , ˆ ρ t g / ) ≤ (cid:126) (cid:90) t g / dt (cid:113) (cid:104) ˆ H ( t ) (cid:105) − (cid:104) ˆ H ( t ) (cid:105) , (C1)where L QF ( ρ , ρ t g / ) = arccos (cid:104)(cid:113) F ( ρ , ρ t g / ) (cid:105) , and F ( ρ , ρ t g / ) = Tr (cid:2)(cid:112) √ ρ ρ t g / √ ρ (cid:3) is the Uhlmann fi-delity. The LHS is the distance of initial and final statesaccording to the quantum Fisher information metric.The RHS of this inequality is the time-integrated in-stantaneous energy uncertainty of our Hamiltonian ˆ H ( t )with the accelerated protocol. Note that the symmetryof our protocol implies the LHS also bounds the energyuncertainty over the interval ( t g / , t g ).To apply this to our accelerated protocol, note thatduring the first half of the evolution, the zero-energy darkstate | d( t ) (cid:105)(cid:104) d( t ) | [Eq. (10)] evolves from the initial state ρ = | ˜1 (cid:105)(cid:104) ˜1 | to an orthogonal state ρ t g / = | a (cid:105)(cid:104) a | . As aresult, the LHS of Eq. C1 becomes π/ ν ( t ), the accelerated Hamiltonian ˆ H ( t ) can be ob-tained from the adiabatic Hamiltonian [Eq. (A3)] bymodifying the original pulse angle and amplitude via [17] θ ( t ) → ˜ θ ( t ) = θ ( t ) + arctan (cid:18) ˙ ν ( t )˙ θ ( t ) / tan[ ν ( t )] (cid:19) , (C2a)Ω → ˜Ω( t ) = (cid:118)(cid:117)(cid:117)(cid:116) ˙ ν ( t ) + (cid:32) ˙ θ ( t )tan[ ν ( t )] (cid:33) . (C2b)Using the fact that the accelerated protocol guaranteesthat the system’s state follows (at all times) the “dressed”dark state | d ν ( t ) (cid:105) defined in Eq. (B1), we can calculatethe instantaneous energy uncertainty (RHS of Eq. (C1))from the probability p ± of the dressed dark state beingin the instantaneous eigenstates | b ± ( t ) (cid:105) of ˆ H ( t ). Here, | b ± ( t ) (cid:105) are the bright states of ˆ H ( t ) with eigenenergies ± ˜Ω / | b ± ( t ) (cid:105) = 1 √ (cid:16) ± sin[˜ θ ( t )] | ˜1 (cid:105) ± e iγ ( t ) cos[˜ θ ( t )] | a (cid:105) + | e (cid:105) (cid:17) . (C3)The probabilities p ± are given by p ± ( t ) = |(cid:104) d ν ( t ) | b ± ( t ) (cid:105)| = 12 (cid:20)(cid:18) ˙ ν ( t )˜Ω ( t ) (cid:19) cos [ ν ( t )] + sin [ ν ( t )] (cid:21) . (C4)Using Eq. (C4), we calculate the instantaneous value of (cid:104) ˆ H ( t ) (cid:105) and (cid:104) ˆ H ( t ) (cid:105) as (cid:104) ˆ H ( t ) (cid:105) = ˜Ω( t )2 ( p + − p − ) = 0 , (C5a) (cid:104) ˆ H ( t ) (cid:105) = [ ˜Ω( t )] p + + p − ) = 14 (cid:40) ˙ ν ( t ) + ˙ θ ( t )sec [ ν ( t )] (cid:41) . (C5b)Substituting Eq. (C5) into Eq. (C1), we then have π ≤ (cid:90) t g / dt (cid:115) ˙ ν ( t ) + ˙ θ ( t )sec [ ν ( t )] ≤ (cid:115)(cid:90) t g / dt (cid:118)(cid:117)(cid:117)(cid:116)(cid:90) t g / dt (cid:32) ˙ ν ( t ) + ˙ θ ( t )1 + tan [ ν ( t )] (cid:33) < t g (cid:118)(cid:117)(cid:117)(cid:116) t g (cid:90) t g / dt (cid:32) ˙ ν ( t ) + ˙ θ ( t )tan [ ν ( t )] (cid:33) = t g RMS . (C6)Note that in going to the second line of Eq. (C6), wehave used the Cauchy-Schwarz inequality and the rela-tion sec ( x ) = 1 + tan ( x ). From Eq. (C6), we can writethe bound for ˜Ω RMS as˜Ω
RMS > π/t g . (C7) Appendix D: Optimal fluxonium parameter regimefor a tripod gate in the T protected regime To get a tripod gate with small non-RWA errors aswell as a qubit with a long T coherence time, we usethe following criteria in choosing the fluxonium circuitparameters:1. The ground states, i.e., low-lying energy levels ofthe tripod which are labeled by | (cid:105) | (cid:105) and | a (cid:105) ,should be well isolated from each other and be non-degenerate. The requirement of strong isolation isnecessary to obtain a T protected qubit. On theother hand, the nondegeneracy of ground states isrequired to ensure sufficiently large detuning of thespurious crosstalk transitions from the driving fre-quencies in order to reduce the coherent errors dueto crosstalks.2. The charge matrix elements coupling the excitedstate to the ground states of the tripod systemshould be large and have the same order of magni-tude. This requirement helps to minimize coherenterrors arising from non-RWA processes.The criterion (1) requires us to choose circuit param-eters that satisfy E J (cid:29) E C and E L (cid:28) E J (for well-localized ground states) as well as 0 < Φ ext ≤ Φ / | e (cid:105) of the tripod gate to be the first excitedstate of the central well that is delocalized over the poten-tial wells where the ground states | (cid:105) , | (cid:105) and | a (cid:105) reside,but yet somewhat separated from the much more densely-spaced higher energy levels. This implies that the state | e (cid:105) must lie in the vicinity of the top edge of the cosinepotential, which requires √ E C E J (cid:29) E J . It is clearthat the criteria (1) and (2) cannot be satisfied simulta-neously in a standard fluxonium circuit. That forces usto seek a balanced parameter set, ensuring that we canboth end up with a long-lived qubit, as well as with atripod that allows for a high-fidelity SATD gate.In order to achieve this balance, we initially do a nu-merical search over experimentally realizable circuit pa-rameters, enforcing conditions on the energy-level struc-ture that do not strongly violate our desired selectioncriteria outlined in points (1) and (2) above. These in-clude, for example, requiring that the charge matrix ele-ment between levels | (cid:105) and | (cid:105) to be small (which max-imizes T ), and that the tripod transition energies ω j e ,with j = 0 , , a, are not degenerate (to minimize effectsof crosstalk), and so on. After applying this procedure,we end up with a much reduced parameter space, whichin turn is used to run a more focused search that opti-mizes over the fidelity of an actual SATD gate (althoughwithout the frequency chirping). Taking all of the aboveinto account, we end up with a final choice of parameters: E L /h = 0 .
063 GHz, E J /h = 9 .
19 GHz, E C /h = 2 GHzand Φ ext = 0 . . We stress that the small inductiveenergy of our fluxonium puts our E L in same regime asthe Blochnium device that was recently realized experi-mentally [49]. The corresponding energy level structurealong with the potential energy landscape, are shown inFig. 1. Due to the small matrix element between thequbit levels | (cid:105) and | (cid:105) ( | n | = 0 . T limited, and due the positioning of the excited level | e (cid:105) ,a tripod with relatively strong tripod matrix elements( | n | = 0 . | n | = 0 .
46 and | n ae | = 0 .
16) can be real-ized.
Appendix E: Accounting for a smooth turn on andoff of the pulse at the beginning and end of theprotocol
Since realistic pulses are off in the beginning and theend of the protocol, we sandwich the pulse in Eq. (8)by a ramp time t ramp during which the pulse ˜Ω ae ( t ) issmoothly turned on (off) at the beginning (end) of theprotocol. The full driving pulses can then be writtenas smooth piecewise continuous functions which can beseparated in three time regions [(I): 0 ≤ t < t ramp , (II): t ramp ≤ t ≤ t g + t ramp , (III): t g + t ramp < t ≤ t g + 2 t ramp )] as˜Ω ( t )Ω = , cos α (cid:40) sin[ θ ( t − )] + cos[ θ ( t − )]¨ θ ( t − )˙ θ ( t − ) + Ω / (cid:41) (II) , , (E1a)˜Ω ( t )Ω = ,e − iβ sin α (cid:40) sin[ θ ( t − )] + cos[ θ ( t − )]¨ θ ( t − )˙ θ ( t − ) + Ω / (cid:41) (II) , , (E1b)˜Ω ae ( t )Ω = P (cid:18) t t ramp (cid:19) (I) ,e − iγt − (cid:40) cos[ θ ( t − )] − sin[ θ ( t − )]¨ θ ( t − )˙ θ ( t − ) + Ω / (cid:41) (II) , − P (cid:18) t − − t g t ramp (cid:19) (III) , (E1c)where t − = t − t ramp . Specifically, we choose t ramp =0 . t g to be short enough compared to the overall pulselength such that it will not significantly affect the wholedynamics but long enough such that there is no sharpjump in the pulse when it is turned on or off. Notethat, in Eq. (E1c) we have used the function P ( x ) givenin Eq. (A6) to ensure a smooth turn on and off of thepulse ˜Ω ae ( t ) during a time duration t ramp at the begin-ning and end of the protocol. The time profile of thedriving pulse with the inclusion of the ramp function isshown in Fig. 7(a). Appendix F: Approximating non-Markovian noisewith Markovian dynamics
In this Appendix, we show that approximating non-Markovian dynamics of 1 /f noise with Markovian masterequation cannot capture the decay rate of all coherencescorrectly. In the main text, we construct our Linbladmaster equation by picking the decay rate Γ k of the co-herences ρ k ≡ (cid:104) k | ˆ ρ | (cid:105) (coherences which involve the ref-erence level, i.e., the qubit state | (cid:105) )) such that the finaldecay of the coherences ρ k , ∀ k (cid:54) = 1 at the gate time t g is the same for our Markovian dynamics as it would befollowing the non-Markovian, Gaussian-lineshape decaywith envelope exp (cid:2) − ( t/T ϕ,kl ) (cid:3) where T ϕ,kl is the freeinduction decay time given by Eq. (26). (Note that wecan equivalently choose the qubit state | (cid:105) as the refer-ence level as both choices will result in the same valueof state-averaged fidelities.) While the decay of ρ k co-herences can be captured correctly, the dephasing timesof coherences that do not involve the reference level ( ρ kl ,7 ∀ k, l (cid:54) = 1) in our Markovian dynamics are in general notthe same as the free induction decay times calculatedfrom Eq. (26). To see this, we can write the Lindbladmaster equation [Eq. (24)] in terms of the density matrixelements as˙ ρ kl ( t ) = − i [ ˆ H ( t ) , ˆ ρ ( t )] kl − (cid:18) sgn (cid:18) ∂ε k ∂ Φ ext (cid:19) (cid:112) Γ k − sgn (cid:18) ∂ε l ∂ Φ ext (cid:19) (cid:112) Γ l (cid:19) ρ kl = − i [ ˆ H ( t ) , ˆ ρ ( t )] kl − t g (cid:16) T eff ϕ,kl (cid:17) ρ kl , (F1)where we have identified1 T eff ϕ,kl ≡ (cid:12)(cid:12)(cid:12)(cid:12) sgn (cid:18) ∂ε k ∂ Φ ext (cid:19) T ϕ,k − sgn (cid:18) ∂ε l ∂ Φ ext (cid:19) T ϕ,l (cid:12)(cid:12)(cid:12)(cid:12) . (F2)We can see that the effective dephasing time T eff ϕ,kl [Eq. (F2)], ∀ k, l (cid:54) = 1 calculated from the Lindblad masterequation is in general not the same as the free induc-tion dephasing time T ϕ,kl calculated from Eq. (26). Infact, as shown in Table I, our approach if anything over-estimates the dominant dephasing processes within thetripod subspace, e.g., T eff ϕ, a0 and T eff ϕ, ae . kl T ϕ,kl ( µ s) T eff ϕ,kl ( µ s)01 7.03 7.03a1 6.97 6.97e1 53.43 53.43a0 3.50 1.75e0 8.09 17.31ae 6.16 3.76TABLE I. Dephasing times for transitions between computa-tional states | k (cid:105) and | l (cid:105) calculated using two different meth-ods: (Middle column): directly from the free induction decayformula [Eq. (26)] and (Right column): indirectly from theLindblad Master equation [Eq. (F2)]. Appendix G: Indirect qubit driving through acoupled cavity
In order to provide Purcell protection, in practice, su-perconducting qubits are often driven indirectly througha coupled cavity. In this Appendix, we discuss the detailsof such driving of our fluxonium qubit, especially in thecontext where the drive power is constrained such thatadditional dissipative mechanisms due to the cavity canbe avoided. In the following, we show how the drivingfield applied to the qubit-coupled cavity is related to thedriving field seen by the qubit V ( t ) [Eq. (2) of the maintext].
1. Driving field for the cavity
In this subsection we derive an explicit relationship be-tween the field that drives the cavity u ( t ) and the voltage V ( t ) [Eq. (2)] seen by the qubit. We begin by writing theHamiltonian for the driven cavity-coupled fluxonium asˆ H JC ( t ) = (cid:88) k ε k | k (cid:105)(cid:104) k | + ω cav ˆ a † ˆ a + (cid:88) k,l gn kl | k (cid:105)(cid:104) l | (ˆ a † + ˆ a )+ u ( t )(ˆ a † + ˆ a ) , (G1)where ε k , | k (cid:105) are the fluxonium eigenenergies and eigen-states, respectively, and ω cav is the cavity frequency. Thequbit-cavity coupling strength is g , while n kl = (cid:104) k | ˆ n | l (cid:105) isthe fluxonium charge matrix element. The driving fieldon the cavity is u ( t ) and the operators ˆ a † and ˆ a arethe cavity photon creation and annihilation operators,respectively.We consider operating the cavity-qubit system in thedispersive regime where gn kl (cid:28) | ε l − ε k − ω cav | , and apply-ing a highly off-resonant drive to limit the cavity photonpopulation. We also consider the drive strength to beweak enough that the cavity photon number is small soto avoid cavity-induced dissipations. In this weak drivingpower regime, we can estimate the relationship betweenthe applied cavity field u ( t ) and the driving field on thequbit V ( t ), by first solving for the classical cavity fieldindependently (i.e., in the limit g → a ( t ) operator can bewritten as [65]˙ˆ a ( t ) = i [ ˆ H JC ( t ) , ˆ a ( t )] − κ a ( t ) − √ κ ˆ b in ( t )= − iω cav ˆ a ( t ) − iu ( t ) − κ a ( t ) − √ κ ˆ b in ( t ) . (G2)Here κ is the cavity photon decay rate (due to the photonleakage to the bath), ˆ b in ( t ) is the standard bath annihila-tion operator which represents the noise [65]. The equa-tion of motion for ˆ a † ( t ) can be obtained by taking theHermitian conjugate of Eq. (G2). By treating the cav-ity field classically and writing the equation of motion interms of the mean value of the photon field displacement x = (cid:104) ˆ a + ˆ a † (cid:105) / √
2, we have¨ x ( t ) = − (cid:18) ω + κ (cid:19) x ( t ) − κ ˙ x ( t ) − √ ω cav u ( t ) , (G3)where we have used (cid:104) ˆ b in ( t ) (cid:105) = (cid:104) ˆ b † in ( t ) (cid:105) = 0. The solutionof Eq. (G3) is x ( t ) = x p ( t ) + x h ( t ) , (G4)where x h ( t ) = Ae − κ t sin( ω cav t + φ ) , (G5)8is the homogeneous solution and x p ( t ) is the inhomoge-neous solution. To solve for x p ( t ), we first define theFourier transform of x ( ω ) and u ( ω ) as x ( ω ) = (cid:90) ∞−∞ x ( t ) e − iωt dt,u ( ω ) = (cid:90) ∞−∞ u ( t ) e − iωt dt, (G6)and their inverse Fourier transform as x ( t ) = 12 π (cid:90) ∞−∞ x ( ω ) e iωt dω,u ( t ) = 12 π (cid:90) ∞−∞ u ( ω ) e iωt dω. (G7)Taking the Fourier transform of Eq. (G3), we have (cid:20) − ω + iκω + ω + κ (cid:21) x ( ω ) = −√ ω cav u ( ω ) , (G8)which gives x ( ω ) = √ ω cav u ( ω ) ω − iκω − ( ω + κ / . (G9)We can relate the mean-photon displacement x ( t ) withthe qubit driving voltage V ( t ) by replacing the cavity de-gree of freedom in the cavity-qubit coupling Hamiltonianby its classical value, i.e.,ˆ H c ≈ (cid:88) kl g (cid:104) ˆ a † + ˆ a (cid:105) n kl | k (cid:105)(cid:104) l | = √ gx ( t ) (cid:88) kl n kl | k (cid:105)(cid:104) l | . (G10)Comparing Eq. (G10) with the qubit driving term in theHamiltonian of Eq. (1), we can identify √ gx ( t ) = V ( t ) , (G11)where in the frequency domain, it can be written as √ gx ( ω ) = (cid:90) ∞−∞ V ( t (cid:48) ) e − iωt (cid:48) dt (cid:48) . (G12)Substituting Eq. (G9) into Eq. (G12), we have u ( ω ) = ( ω − iκω − (cid:0) ω + κ / (cid:1) )2 gω cav (cid:90) ∞−∞ V ( t (cid:48) ) e − iωt (cid:48) dt (cid:48) . (G13)Finally, taking the inverse Fourier transform, we arriveat an expression for the time-domain form of the cavitypulse u ( t ) that is required to generate a qubit drive V ( t )as u ( t ) = − gω cav (cid:20) d dt + κ ddt + ω + κ (cid:21) V ( t ) . (G14)
2. Relation between the RMS voltage and thetime-averaged cavity photon number
In this section, we establish the relationship betweenthe RMS voltage ˜ V RMS of the qubit driving field and thetime-averaged cavity photon number ¯ n cav where¯ n cav = 1 t g (cid:90) t g (cid:104) ˆ a † ˆ a (cid:105) dt ≈ t g (cid:90) t g (cid:104) ˆ a † (cid:105)(cid:104) ˆ a (cid:105) dt = 1 t g (cid:90) t g dt | η ( t ) | . (G15)Here, we treat the photon field classically, which allows usto replace (cid:104) ˆ a † (cid:105) and (cid:104) ˆ a † (cid:105) by the classical field amplitudes η ∗ ( t ) and η ( t ), respectively. Using this classical field inEq. (G10), we can identify the qubit driving voltage as V ( t ) = 2 g Re[ η ( t )] . (G16)The RMS voltage of the qubit driving field is then givenby ˜ V RMS = (cid:115) t g (cid:90) t g | V ( t ) | = g √ n cav , (G17)where in evaluating the second line, we have used therelation1 t g (cid:90) t g [Re( η ( t ))] = (cid:88) j t g (cid:90) t g | η j ( t ) | cos ( ω j t + φ j ) (cid:39) t g (cid:90) t g | η ( t ) | = ¯ n cav . (G18)Without loss of generality, in Eq. (G18), we have writ-ten the photon field as a multicomponent field, i.e., η ( t ) = (cid:80) j η j e − i ( ω j t + φ j ) . To avoid cavity-induced dis-sipation mechanisms, it is preferable to have a smallcavity photon number [58, 60], i.e., ¯ n cav = (cid:104) ˆ a † ˆ a (cid:105) (cid:28) n cav = 0 .
05. This constrains the maximum RMS voltagefor a fixed cavity-qubit coupling strength g as shown inEq. (32) of the main text. Each vertical cut of the plotin Fig. 11(b) corresponds to different maximum allowedvalues of ˜ V RMS where the purple vertical lines show ex-plicitly two different fixed values of the cavity-qubit cou-pling strength g corresponding to two different maximumallowed values of ˜ V RMS .
3. Effects of the cavity on qubit’s T and T times In this section we briefly outline the effects of the cou-pled cavity on the coherence times of the qubit, andshow that they do not limit the performance of ourgates. In particular, we consider a regime where thecavity photon decay rate κ (cid:28) g / | ∆ cav kl, ± | , ∀ k, l where9∆ cav kl, ± = ε l − ε k ± ω cav is the detuning of the cavity fre-quency ω cav from the | k (cid:105) ↔ | l (cid:105) energy transition of thequbit. The first noise channel we consider is the Purcellrelaxation time, which we denote as T , cav and can becalculated as [66]( T , cav ) jk = (∆ cav kl, ± ) / ( κg | n kl | ) . (G19)Since the all the relevant transitions are highly detunedfrom the cavity frequency, we expect this kind of pro-cess to be weak. Another relevant noise channel is dueto photon shot noise, and leads to pure dephasing. Wedenote the corresponding time scale as T , cav , which canbe approximated by [67, 68] T , cav = 1 / ( κ ¯ n cav ) . (G20) We stress that Eq. (G20) is valid in the limit of smallcavity photon decay relative to all the relevant disper-sive shifts ( κ (cid:28) g / | ∆ cav kl, ± | ), a regime that we are in-terested in. Moreover, we also consider operating in thedispersive limit ( gn kl (cid:28) | ∆ cav kl, ± | , ∀ k, l ) and small-cavity-photon-number regime (¯ n cav (cid:28)
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15s and T , cav ≈ .
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