Enhanced Quantum State Transfer and Bell State Generation over Long-Range Multimode Interconnects via Superadiabatic Transitionless Driving

Modular design of quantum computers Devitt_Architectural_2009 ; Monroe_Large_2014 ; Bravyi_Future_2022 relaxes wiring and control complexity, as well as cryogenic cooling power requirements, of the underlying Quantum Processing Units (QPU), and is the path forward for the required scaling Fowler_High_2009 ; Fowler_Surface_2012 ; Bravyi_High_2023 towards quantum error correction Knill_Theory_1997 ; Gottesman_Stabilizer_1997 ; Knill_Theory_2000 ; Lidar_Quantum_2013 ; Terhal_Quantum_2015 . For superconducting qubits, this vision necessitates developing interconnects at various levels of modularity Bravyi_Future_2022 , such as dense short-range interconnects Gold_Entanglement_2021 ; Conner_Superconducting_2021 to extend the effective size of QPUs, and sparse meter-range interconnects to enable parallelization of multiple QPUs within a dilution fridge. The short-range interconnect length is comparable to the distance between the qubits within a single chip and behaves effectively as a single mode system. While standard two-qubit gates could potentially work across a short interconnect, the multimode nature of long-range interconnects makes direct two-qubit gates more difficult. Two alternatives are to use the interconnect to perform state transfer, or to generate remote entanglement such as a Bell state. In conjunction with local operations and classical communication, either of these operations can be used as a resource to implement indirect remote two-qubit gates Gottesman_Demonstrating_1999 ; Eisert_Optimal_2000 ; Huang_Experimental_2004 .

The past few years have seen a recent surge into numerous superconducting circuit realizations of remote entanglement generation and quantum state transfer Kurpiers_Deterministic_2018 ; Axline_Demand_2018 ; Campagne_Deterministic_2018 ; Leung_Deterministic_2019 ; Zhong_Violating_2019 ; Bienfait_Phonon_2019 ; Chang_Remote_2020 ; Zhong_Deterministic_2021 ; Yan_entanglement_2022 ; Kannan_Demand_2023 ; Niu_Low_2023 ; Qiu_Deterministic_2023 . These protocols can be broadly categorized as either employing time-symmetric emission and capture of itinerant photons Kurpiers_Deterministic_2018 ; Axline_Demand_2018 ; Campagne_Deterministic_2018 ; Zhong_Violating_2019 ; Bienfait_Phonon_2019 ; Kannan_Demand_2023 ; Niu_Low_2023 ; Qiu_Deterministic_2023 , or using qubit interactions with the standing-wave modes of meter-long scale interconnects Leung_Deterministic_2019 ; Zhong_Violating_2019 ; Chang_Remote_2020 ; Zhong_Deterministic_2021 ; Qiu_Deterministic_2023 . Among protocols based on standing-wave modes, Stimulated Raman Adiabatic Passage (STIRAP) Gaubatz_Population_1990 ; Vitanov_Stimulated_2017 ; Bergmann_Roadmap_2019 achieves better fidelity Chang_Remote_2020 compared to a qubit-interconnect-qubit direct excitation exchange, also referred to as the relay protocol Zhong_Violating_2019 , by protection against interconnect loss.

In STIRAP Gaubatz_Population_1990 ; Vitanov_Stimulated_2017 ; Bergmann_Roadmap_2019 , we evolve the dark eigenstate of a Lambda system adiabatically towards a desired target state, applicable to quantum state transfer and entanglement generation. One advantage is the suppression of potential relaxation through the intermediate lossy interconnect. The operation speed is, however, limited by leakage to the bright lossy eigenstates whose transition frequency is set by the coupling strength. Transitionless Driving (TD) methods Demirplak_Adiabatic_2003 ; Demirplak2008consistency ; Berry_Transitionless_2009 ; Ibanez_Multiple_2012 ; Zheng_Optimal_2022 cancel out non-adiabatic transitions exactly via a modified control Hamiltonian, similar in spirit to the perturbative Derivative Removal by Adiabatic Gate (DRAG) technique Motzoi_Simple_2009 ; Gambetta_Analytic_2011 ; Malekakhlagh_Mitigating_2022 ; Li_Suppression_2023 . One potential practical drawback can however be the need for a control knob not accessible by the original Hamiltonian. SuperAdiabatic Transition Driving (SATD) Baksic_Speeding_2016 ; Zhou_Accelerated_2017 redefines the evolution path, connecting the original source and target states, such that in the dressed frame the non-adiabatic transitions are canceled out exactly without the need for additional control knobs. SATD solutions for STIRAP have also been generalized to single-qubit tripod gates Ribeiro_Accelerated_2019 ; Setiawan_Analytic_2021 and more recently to two-qubit gates Setiawan_Fast_2023 for fluxonium qubits Manucharyan_Fluxonium_2009 .

In this paper, we characterize the performance improvements of the SATD protocol against STIRAP, and promote its usage for quantum state transfer and Bell state generation in a multimode interconnect setting. By removing leakage to the resonant interconnect mode, the operation speed for SATD is not limited by the qubit-interconnect coupling $g$ anymore, but determined by the interconnect Free Spectral Range (FSR) $\Delta_{c}$, leading to a significant speedup as well as a robustness to variation in $g$ compared to STIRAP. We show that the single-mode SATD solutions work reasonably well for a multimode interconnect with sufficiently large FSR ($\Delta_{c}\gg g$), and quantify the deviations from expected behavior due to multimode effects. In particular, we find that the even-odd mode dependence of the interaction breaks the dark-state symmetry, which is in principle detrimental to such dark-state-based adiabatic protocols. This impacts the Bell state generation more by an adiabatic overlap error proportional to $(g/\Delta_{c})^{2}$, and can be mitigated only via a weaker $g$. This weaker $g$ requirement due to multimode effects, and the $g$ robustness of SATD makes its application very advantageous especially for Bell state generation. Furthermore, we observe improvements by SATD in suppressing the incoherent error due to qubits relaxation, pure dephasing, and the interconnect quality factor.

The remainder of this work is organized as follows. Section II describes the system under consideration with two quantum modules connected via a multimode interconnect, and a Lindblad model introduced for our analytical and numerical analyses. In Sec. III, we revisit the ideal single-mode STIRAP protocol, used for quantum state transfer and entanglement generation, and discuss potential detrimental multimode sources of error. In Sec. IV, we present extensive simulations investigating the numerous advantages of SATD compared to regular STIRAP in such a multimode context. We further assess the performance of indirect two-qubit gates achieved by combining quantum state transfer and remote entanglement with local operations. Appendix A discusses the details of our Lindblad model and simulations, and provides a numerical convergence test. Appendices B and C review the single-mode STIRAP, and complications that arise due to a multimode interconnect, respectively. In Appendix D, we review the derivation of SATD solutions for single-mode STIRAP following Ref. Baksic_Speeding_2016 .

We consider a system consisting of two tunable-frequency qubits that have tunable interactions to a common long-range multimode interconnect as depicted in Fig. 1(a). The standard motivation for such a setup is to perform remote quantum operations between two modules (chips) connected via a long interconnect such as a coaxial cable. This can, however, be also relevant to on-chip transmission lines between distant qubits Sundaresan_Beyond_2015 . Notable experimental studies have employed tunable grounded Transmon qubits with tunable RF SQUID couplers Barends_Coherent_2013 ; Chen_Qubit_2014 ; Geller_Tunable_2015 connected via an on-chip transmission line Chang_Remote_2020 or a cable Qiu_Deterministic_2023 . Although this work is motivated by superconducting architectures, the following analysis and characterization of remote operations is presented in a system-agnostic manner.

We characterize the performance of STIRAP and SATD protocols for quantum state transfer and Bell state generation via a Lindblad simulation that accounts for qubit relaxation ($T_{1}$), pure dephasing ($T_{2\phi}$) and cable relaxation ($\kappa_{n}$):

where $\hat{\rho}_{s}(t)$ is the system density matrix, and $\mathcal{D}[\hat{C}]\hat{\rho}_{s}\equiv\hat{C}\hat{\rho}_{s}\hat{C}^{{\dagger}}-(1/2)\{\hat{C}^{{\dagger}}\hat{C},\hat{\rho}_{s}\}$ is the dissipator for the collapse operator $\hat{C}$. We model the qubits as weakly anharmonic, and the interconnect as a collection of harmonic quantum oscillators, with time-dependent (controllable) qubit-interconnect interaction as:

where $\omega_{q}(t)$, $\alpha_{q}$, $\omega_{n}=\omega_{n_{c}}+n\Delta_{c}$ and $g_{qn}(t)$ are the qubit frequency, anharmonicity, evenly spaced $n^{th}$ mode frequency with FSR $\Delta_{c}$, and qubit-interconnect interaction rates for $q=a,b$, respectively. Moreover, $n_{c}$ is the center mode index, and $N$ is the additional modes kept on each side. In writing Hamiltonian (2), we have made certain approximations, motivated by the physics of multimode interconnects (see Appendix A), similar to Ref. Chang_Remote_2020 . An important feature of Hamiltonian (2) is the even-odd mode-dependent relative sign for the qubit-interconnect interaction, which accounts for the distinct spatial profile of even and odd interconnect modes Chang_Remote_2020 .

STIRAP is a protocol for adiabatic transfer of population in a Lambda system, i.e. between two quantum states coupled through a common intermediate state, via temporal control of the interactions [Fig. 1(b)]. Under the single-mode (ideal) case, the Hamiltonian reads

where we assume all levels are resonant. This resonant Lambda system has a dark eigenstate

and two bright eigenstates

with the mixing angle defined as $\tan\theta(t)\equiv g_{ac}(t)/g_{bc}(t)$.

The dark eigenstate, having no overlap with the intermediate (possibly) lossy interconnect state, therefore allows for an adiabatic quantum state transfer by arbitrarily evolving the mixing angle $\theta(t)$. A common choice for the controls are $g_{ac}(t)=g\sin\theta(t)$ and $g_{bc}(t)=g\cos\theta(t)$ with $\theta(t)=(t/\tau_{p})\theta_{p}$ for $t\in[0,\tau_{p}]$. For instance, sweeping $\theta(t)$ from 0 to $\pi/2$ or $\pi/4$ should ideally implement $\ket{1_{a}0_{c}0_{b}}\rightarrow-\ket{0_{a}0_{c}1_{b}}$ (full state transfer), or $\ket{1_{a}0_{c}0_{b}}\rightarrow 1/\sqrt{2}(\ket{1_{a}0_{c}0_{b}}-\ket{0_{a}0_{c}1_{b}})$ (Bell state), respectively. Note, however, that unwanted non-adiabatic $\ket{D(t)}\rightarrow\ket{B_{\pm}(t)}$ transitions, whose probabilities grow with $\dot{\theta}(t)$, impose a limit on the STIRAP speed (Fig. 1(c) and Appendix B).

For a multimode interconnect, with the interaction $g$ comparable to the FSR $\Delta_{c}$, the adjacent modes impact the fidelity of STIRAP detrimentally by (i) breaking the dark-symmetry condition, (ii) introducing additional leakage, and (iii) additional decay channels. Regarding item (i), each adjacent mode forms an effective off-resonant Lambda system with the qubits [Fig. 1(b)]. Our extended multimode STIRAP analysis suggests that for a hypothetical interconnect with same-sign interactions the Hamiltonian supports the original dark eigenstate, while for the physical case of mode-dependent interactions, one instead finds a pseudo dark eigenstate having a non-zero overlap $|[g(t)/\Delta_{c}]\sin[2\theta(t)]|$ with the one-excitation subspace of the odd (opposite-sign) modes (Appendix C). Such an adiabatic error vanishes for quantum state transfer with $\theta(\tau_{p})=\pi/2$, but is maximized for Bell state generation with $\theta(\tau_{p})=\pi/4$ requiring a weaker $g$ for better fidelity.

Figure 2 characterizes the performance of STIRAP in such a multimode context, where we simulate the Lindblad Eqs. (1)–(2) numerically with five interconnect modes evenly spaced about the qubit frequency for the common sine/cosine STIRAP controls and initial pure state $\hat{\rho}_{s}(0)\equiv\ket{1_{a}0_{c}0_{b}}\bra{1_{a}0_{c}0_{b}}$. We define error,

in terms of the overlap of the final density matrix with the ideal target states for state transfer and Bell states as $\ket{\psi_{\text{id}}}=\ket{0_{a}0_{c}1_{b}}$ and $\ket{\psi_{\text{id}}}=(1/\sqrt{2})(\ket{1_{a}0_{c}0_{b}}-\ket{0_{a}0_{c}1_{b}})$, respectively. Panels (a) and (b) show the corresponding error as a function of $g$ for fixed FSR of $\Delta_{c}/2\pi=100$ MHz and various interconnect quality factors, where we observe distinct approximate optimal $g/\Delta_{c}$ ratio of $15\%$ and $4\%$ for state transfer and Bell state, respectively, at $Q_{c}=10^{5}$ and $T_{1a}=T_{1b}=100\ \mu$s. The optimal couplings are a balance between more leakage to the neighboring modes at stronger $g$ (faster operation) and more incoherent error at weaker $g$ (slower operation). The Bell state generation also suffers from a non-zero adiabatic overlap with the odd interconnect modes due to dark-state symmetry breakdown (Appendix C and Sec. IV.2). Using the optimal couplings in panel (c), we find that in order to reach sub-percent error the interconnect $Q_{c}$ must approximately exceed $2.2\times 10^{4}$ and $6.5\times 10^{4}$ for state transfer and Bell state, respectively.

TD is a control technique for cancelling out non-adiabatic transitions via a corrected control Hamiltonian Demirplak_Adiabatic_2003 ; Demirplak2008consistency ; Berry_Transitionless_2009 ; Ibanez_Multiple_2012 . In an ideal case, from the hardware perspective, the correction can be simply implemented via a modification of the original control pulses. The superadiabatic aspect refers to implementing the cancellation in a dressed frame, i.e. effectively evolving the initial state in a modified path in the Hilbert space towards the target state. The standard single-mode STIRAP problem allows for a family of exact SATD solutions Baksic_Speeding_2016 . A commonly employed SATD solution dresses the evolution path along the spin-1 $\hat{M}_{x}$ operator, yielding the explicit results Baksic_Speeding_2016 ; Ribeiro_Accelerated_2019 ; Setiawan_Analytic_2021 ; Setiawan_Fast_2023 (see also Appendix D)

where the corrections depend on both the first $\dot{\theta}(t)$ and the second derivative $\ddot{\theta}(t)$ of the mixing angle (check Appendix D for a comparison of pulse shapes).

In the following, we characterize and compare the performance of regular STIRAP and SATD protocols, and dissect various favorable aspects of SATD usage in the context of multimode interconnects for quantum state transfer and Bell state generation. Some advantages of the SATD protocol can be summarized as follows:

• (i)

  The speed of standard STIRAP is limited by the dark$\rightarrow$bright transitions, whose effective transition frequency is equivalent to the resonant interaction rate $g$ (Appendix B). SATD, however, removes the dark$\rightarrow$bright leakage, and allows for faster operations whose speed limit is set by leakage to the adjacent interconnect modes (Appendix D, Figs. 3 and 7).

• (ii)

  One crucial practical consequence of (i) is the SATD robustness to qubit-interconnect interaction $g$, and the possibility of performing fast high-fidelity operations even with weak interactions (Fig. 4).

• (iii)

  SATD provides a more pronounced speedup over STIRAP for the Bell state generation. This is the case as Bell state generation is more sensitive to the even/odd sign dependence of the interaction, compared to state transfer, and requires a weaker coupling to mitigate adiabatic overlap with the odd modes (Appendix C and Fig. 7).

• (iv)

  We also observe improved sensitivity of the SATD protocol error, compared to STIRAP, with respect to qubit and interconnect mode relaxation, as well as qubit pure dephasing (Figs. 5–6 and 8–9).

In this work, we promote the general application of shortcuts to adiabaticity methods Jarzynski_Generating_2013 ; Deffner_Classical_2014 ; Guery_Shortcuts_2019 , in particular SATD for STIRAP Baksic_Speeding_2016 ; Ribeiro_Accelerated_2019 ; Setiawan_Analytic_2021 ; Setiawan_Fast_2023 , in improving remote entanglement generation and quantum state transfer in multimode interconnects. Our results have applications to both long-range QPU-QPU and potential on-chip connections. Besides introducing new leakage and decay channels, we find the multimode nature of an interconnect violates the dark state symmetry required for adiabatic passage by an adiabatic overlap error with the odd modes that grows as $(g/\Delta_{c})^{2}$, which impacts entanglement generation more strongly. This observation makes SATD a great fit in this multimode context, since due to its robustness against $g$ it allows for fast quantum operations at a sufficiently weak $g$ that suppresses the overlap error as well. For a meter-long interconnect with FSR of 100 MHz, we can calibrate $\mathcal{O}(50)$ ns quantum state transfer and Bell state with sub-percent error.

We find the single-interconnect-mode SATD solutions Baksic_Speeding_2016 ; Ribeiro_Accelerated_2019 ; Setiawan_Analytic_2021 ; Setiawan_Fast_2023 to work approximately as intended in the weak $g$ limit such that only leakage in the resonant subspace (dark-bright transitions) is cancelled out, and the operation speed is limited by leakage to the adjacent modes, which is set by the interconnect FSR. A potential future research direction is expediting the operation even further by suppressing leakage to the off-resonant modes. One could ask whether precise or approximate SATD solutions exist in the multimode case, and whether they can be implemented via the same control knobs. Also, the single-mode SATD solutions can potentially serve as a reasonable initial guess for numerical optimal control techniques Khaneja_Optimal_2005 ; Koch_Quantum_2022 for further leakage improvement.

We appreciate helpful discussions with the IBM Quantum members Vikesh Siddhu, Theodore J. Yoder, Alireza Seif, Luke C. G. Govia, Muir Kumph, Jerry M. Chow, and Jay M. Gambetta. The authors acknowledge the IBM Research Cognitive Computing Cluster (CCC) service for providing resources that have contributed to the research results reported within this paper.