# Quantum Simulation and Optimization in Hot Quantum Networks

###### Abstract

We propose and analyze a setup based on (solid-state) qubits coupled to a common multi-mode transmission line, which allows for coherent spin-spin interactions over macroscopic on-chip distances, without any ground-state cooling requirements for the data bus. Our approach allows for the realization of fast deterministic quantum gates between distant qubits, the simulation of quantum spin models with engineered (long-range) interactions, and provides a flexible architecture for the implementation of quantum approximate optimization algorithms.

Introduction.—One of the leading approaches for scaling up quantum information systems involves a modular architecture that makes use of a combination of short and long-distant interactions between the qubits monroe16 (); vandersypen17 (). In particular, long-distant interactions can be implemented via a quantum bus which can effectively distribute quantum information between remote qubits, as shown in the context of of trapped ions Poyatos1998 (); Molmer1999 (); milburn99 (); Ripoll2005 (); lemmer13 (), solid state systems schuetz17 (); Royer2017 (), electromechanical resonators rabl10 (), as well as circuit QED architectures scarlino18 (); woerkom18 (); Gambetta2017 (); wendin17 (); hanson07 (); zwanenburg13 (). In this Letter, we provide a unified theoretical framework for robust distribution of quantum information via a quantum bus that operates at finite temperature temperature (), fully accounts for the multi-mode structure of the data bus, and does not require the qubits to be identical. Our approach [c.f Fig. 1(a)] results in an architecture where fully programmable interactions between qubits can be realized in a fast and deterministic way, without any ground-state cooling requirements for the data bus, thereby setting the stage for various applications in the context of quantum information processing Northup2014 () in a hot quantum network, different from quantum state transfer discussed previously cirac97 (); Vermersch2017 (); Xiang2017 (). As illustrated in Fig. 1(b), and discussed in detail below, one can use our scheme to deterministically implement (hot) quantum gates between two qubits. Moreover, we present a recipe to generate a targeted and scalable evolution for a large set of qubits coupled via a single transmission line, thereby providing a natural architecture for the implementation of quantum algorithms, such as quantum annealing Das2008 () or the quantum approximate optimization algorithm (QAOA) farhi14 (); farhi16 (); otterback17 (), designed to find approximate solutions to hard, combinatorial search problems.

The model.—We consider a set of qubits with corresponding transition frequencies (typically in the microwave regime) that are coupled to a (multi-mode) transmission line of length ; compare Fig. 1 for a schematic illustration. The transmission line is described in terms of photonic modes with wave-vectors , with a linear spectrum , where is the frequency of the fundamental mode and is the (effective) speed of light. As opposed to transversal (Jaynes-Cummings-like) spin-resonator coupling, here we focus on longitudinal coupling as could be realized (for example) with superconducting qubits Kerman2013 (); Billangeon2015 (); Didier2015 (); Richer2016 (); Royer2017 () or quantum dot based qubits childress04 (); harvey18 (); schuetz17 (); Royer2017 (); jin12 (); beaudoin16 (); russ17 (). The Hamiltonian under consideration then reads ()

(1) |

with the Pauli matrices describing the qubits and the coupling strength between qubit and mode . We show below that for specific times , which are integer multiples of the round-trip time , the dynamics of the qubits and all photons fully decouple, while giving rise to an effective interaction between the qubits.

Analytical solution of time evolution.—With the help of the spin-dependent, multi-mode displacement transformation , in our model the spin dynamics can be decoupled from the resonator dynamics (in the polaron frame), and we find , where

(2) |

with the effective spin-spin interaction

(3) |

Therefore, the corresponding time-evolution in the lab frame reads . Consider now the evolution at stroboscopic times ( positive integer), corresponding to multiples of the round trip time . In this case, the synchronization of the modes implies that the full evolution in the lab frame reduces exactly to ,

(4) |

Accordingly, for certain times the qubits fully disentangle from the (thermally populated) resonator modes, thereby providing a qubit gate that is insensitive to the state of the resonator, while imposing no conditions on the qubit frequencies . For specific times, the time evolution in the polaron and the laboratory frame coincide and fully decouple from the photon modes, allowing for the realization of a thermally robust gate, without any need of cooling the transmission line to the vacuum schuetz17 (). Moreover, our approach can be straightforwardly combined with standard spin-echo techniques in order to cancel out efficiently low-frequency noise: By synchronizing fast global rotations with the stroboscopic times , one can enhance the qubit’s coherence times from the time-ensemble-averaged dephasing time to the prolonged timescale .

Frequency cutoff.—In principle, the spin-spin coupling strength as defined in Eq. (3) involves all modes , naively leading to unphysical divergencies, as discussed in the context of transversal qubit-resonator coupling in Refs. filipp11 (); houck08 (). In any physical implementation, however, there is a microscopic lengthscale that naturally introduces a frequency cutoff. Specifically, we take the coupling parameters as , to account for the fact that the qubits couple to the local voltage, where accounts for the microscopic spatial extension of the qubit-transmission line coupling (cf. SM () for details); the factor derives from the scaling of the rms zero-point voltage fluctuations with the mode index , which also implies . In the examples below, we will consider for simplicity a box function , leading to . Note that if the microscopic lengthscale is set to zero, yielding the (point-like) standard result sundaresan15 (), the summation over in Eq. (3) does not converge. Instead for a finite , and for the effective spin-spin interaction Eq. (3) simplifies to (c.f. SM ()). Accordingly, within this exemplary model, the effective coupling does not depend on the microscopic lengthscale , nor the position of the qubits , and scales as , as the rate at which interactions between qubits are generated is limited by the propagation time () of light through the waveguide.

Applications.—In what follows, we discuss three applications of our scheme, with a gradual increase in complexity, namely (i) a hot two-qubit phase gate, (ii) the engineering of spin models, and (iii) the implementation of QAOA in the presence of decoherence and finite temperature. To this end, we consider the possibility to potentially boost and fine-tune the effective spin-spin interactions by parametrically modulating the longitudinal spin-resonator coupling, as could be realized with both superconducting qubits Royer2017 () or quantum dot based qubits harvey18 (); cf. SM () for further details.

Hot phase gate.—As a first illustration of our scheme, we consider the realization of a phase-gate between two remote qubits , placed at each edge of the transmission line (, ). Our initial state consists of a pure initial qubit state with and a thermal state of the waveguide with , and we use Matrix-Product-States (MPS) techniques Peropadre2013 () to show numerically how the hot quantum network generates the desired evolution Eq. (4). We fix which (under ideal circumstances) leads to a maximally entangled pure state after one round trip time (generalizations thereof are provided in SM ()). In Fig. 1(b), we show the von-Neumann entropy and the concurrence of the two-qubit density matrix , showing the realization of the gate at , in the presence of thermal occupation of the waveguide. The corresponding fidelity defined as overlap of with respect to the ideal state is shown in Fig. 2(a). In panels (b) and (c) both the mode occupation and the real space occupation are displayed, with , , referring to the discrete sine transform of . At the round trip time , the waveguide returns to its initial thermal state, as expected. In panel (d), we study the scaling of timing errors by showing the evolution of the error around . In the limit of small errors, the numerical results are well approximated by (black line), with . Accordingly, the timing error is sensitive to the cutoff (as it controls the frequency scale of the couplings), and scales linearly with the effective spin-spin interaction , as slower dynamics are less vulnerable to timing inaccuracies ; for further details, in particular related to the influence of temperature on timing errors, and effects due to nonlinear dispersion relations , cf. SM ().

Engineering of spin models.—We now extend our discussion to the multi-qubit case and provide a recipe how to generate a targeted and scalable unitary with desired spin-spin interaction parameters . To this end, we consider a sequence of successive cycles where for each stroboscopic cycle (labeled by ) we may apply different coupling amplitudes, i.e., . For example, this could be done by pulsing the amplitudes via microwave control harvey18 (); Royer2017 (). The evolution at the end of the sequence is then given by , with and being the total run time. A straightforward way to generate the desired unitary, i.e., to obtain , consists in diagonalizing the target matrix as in terms of real eigenvalues and real eigenstates . This leads immediately to the condition to generate exactly within number of cycles, with , where denotes the largest available spin-spin coupling eigenvalue (). In other words, we can engineer efficiently arbitrary spin-spin interactions after a time which only scales linearly with the number of qubits; in the presence of spin echo. These aspects are illustrated in Fig. 3, where we provide examples for and both (a) a 1D long-range spin model with power law decay () and (b) a 2D model with nearest neighbor interactions (NN). The latter demonstrates that our recipe allows for the realization of general spin models in any spatial dimension and geometry (using a simple one-dimensional physical setup). For both models, we observe the progressive emergence of the target spin interaction with increasing values for , reaching the exact matrix at . The case of a spin glass with random interactions, and the convergence analysis with respect to are presented in SM ().

QAOA.—Finally, we show how to generalize the techniques outlined above in order to implement quantum algorithms that provide approximate solutions for hard combinatorial optimization problems such as Max-Cut [c.f. Fig. 4 and SM ()]. As shown in Refs.farhi14 (); farhi16 (), good approximate solutions to these kind of problems can be found by preparing the state , with , and , where is the cost Hamiltonian encoding the optimization problem, starting initially from a product of eigenstates, i.e., , with . In our scheme, this family of states can be prepared by alternating single-qubit operations ) with targeted spin-spin interactions generated as described above, with . Accordingly, for QAOA we repeat our spin-engineering recipe -times with single-qubit rotations interspersed in between. This preparation step is then followed by a measurement in the computational basis, giving a classical string , with which one can evaluate the objective function of the underlying combinatorial problem at hand. Repeating this procedure will provide an optimized string , with the quality of the result improving as the depth of the quantum circuit is increased farhi14 (); farhi16 (). To illustrate and verify this approach, we have numerically simulated QAOA with up to qubits solving Max-Cut for several -regular graphs with weights , as depicted in Fig. 4(a) and Fig. 1(c), based on our model Hamiltonian given in Eq.(1), while accounting for both finite temperature and decoherence in the form of qubit dephasing and rethermalization of the resonator mode. While our general multi-mode setup should (in principle) be well suited for the implementation of QAOA, here (in order to allow for an exact numerical treatment) we consider a simplified single-mode problem (with resonator frequency ), as could be realized using the resonance condition introduced by a monochromatically modulated coupling harvey18 (); Royer2017 (). Specifically, we simulate the Hamiltonian with controllable couplings harvey18 (); Royer2017 (), detuning and , supplemented by standard dissipators to account for (i) qubit dephasing on a timescale and (ii) rethermalization of the resonator mode with an effective decay rate T1decay (); cf. SM () for further details. As demonstrated in Fig. 1(c), for small-scale quantum systems (that are accessible to our exact numerical treatment) our protocol efficiently solves Max-Cut with a circuit depth of , finding the ground-state energy with very high accuracy (blue curve), corresponding to 4 cuts (shown in red in the inset), even in the presence of moderate noise [compare the cross and plus symbols in Fig. 1(c)].

Decoherence and implementation.—Based on our numerical findings and further analytical arguments, we now turn to the eventual limitations imposed by decoherence. Here, we focus on the QAOA protocol, since both our (i) hot gate (cf. SM () for a full decoherence-induced error analysis thereof) and (ii) the spin engineering protocol can be viewed as less demanding limits of QAOA, where either or (or both) are small, thereby yielding comparatively smaller errors because of a shorter run-time; for example, for the two-qubit phase gate , . The total QAOA run-time can be upper-bounded as , with and the factor corresponding to the (maximum) time required to implement all eigenvalues of the Max-Cut problem. To keep decoherence effects minimal, this timescale should be shorter than all relevant noise processes. The accumulated dephasing-induced error can be estimated as , where is the effective many-body dephasing rate (c.f. SM ()); as shown in Fig. 4(c), we have numerically confirmed this scaling for all graphs shown in panels Fig. 4(a) and Fig. 1(c). Similarly, as demonstrated in Fig. 4(d), the indirect rethermalization-induced dephasing error, mediated by incoherent evolution of the resonator mode, can be quantified as , with total linewidth . The total decoherence-induced error can then be optimized with respect to , yielding the compact expression , with the cooperativity . With this expression, we can bound the maximum number of qubits and circuit depth for a given physical setup with cooperativity .

Specifically, our scheme could be implemented based on superconducting qubits or quantum-dot based qubits coupled by a common high-quality transmission line, with details given in SM (). For concreteness, let us consider quantum-dot based qubits beaudoin16 (); jin12 (); schuetz17 (); harvey18 (); russ17 () where longitudinal coupling could be modulated via both the detuning harvey18 () or inter-dot tunneling parameter jin12 (), respectively. With projected two-qubit gate times of harvey18 (); jin12 (), a coherence time of veldhorst14 (); veldhorst15 (), and with quality factor barends08 (); megrant12 (); bruno15 (), we estimate decoherence errors to be small () for up to qubits and a QAOA circuit depth of for a graph with , respectively, even in the presence of non-zero thermal occupation with . A similar analysis can be made for superconducting qubits SM (). Note that these estimates might be very conservative, as the essential figure of merit in QAOA is not the quantum state fidelity but the probability to find the optimal (classical) bit-string in a sample of projective measurements , which are obtained after many repetitions of the experiments.

Conclusion.—To conclude, we have presented a protocol to generate fast, coherent, long-distance coupling between solid-state qubits, without any ground-state cooling requirements. While this approach has direct applications in terms of the engineering of spin models — e.g. to implement quantum optimization algorithms — it would be interesting to further develop our theoretical treatment in order to increase the level of robustness of our scheme, e.g. to apply protocols based on error correcting photonic codes Michael2016 (), which can protect against single photon losses or rethermalization. Yet another interesting research direction would be to adapt our scheme to other physical setups, say solid-state defect centers coupled by phonons rabl10 ().

###### Acknowledgements.

Acknowledgments.—We thank Shannon Harvey, Hannes Pichler, Pasquale Scarlino, Denis Vasilyev, Shengtao Wang and Leo Zhou for fruitful discussions. Numerical simulations were performed using the ITensor library (http://itensor.org) and QuTiP Johansson2013 (). MJAS would like to thank the Humboldt foundation for financial support. LMKV acknowledges support by an ERC Synergy grant (QC-Lab). JIC acknowledges the ERC Advanced Grant QENOCOBA under the EU Horizon 2020 program (grant agreement 742102). Work in Innsbruck is supported by the ERC Synergy Grant UQUAM, the SFB FoQuS (FWF Project No. F4016-N23), and the Army Research Laboratory Center for Distributed Quantum Information via the project SciNet. Work at Harvard University was supported by NSF, Center for Ultracold Atoms, CIQM, Vannevar Bush Fellowship, AFOSR MURI and Max Planck Harvard Research Center for Quantum Optics. M.J.A.S. and B.V. contributed equally to this work.## References

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Supplemental Material for:

Quantum Simulation and Optimization in Hot Quantum Networks

M.J.A. Schuetz, B. Vermersch, G. Kirchmair, L.M.K. Vandersypen, J.I. Cirac, M.D. Lukin, and P. Zoller

Physics Department, Harvard University, Cambridge, MA 02318,USA

Center for Quantum Physics, and Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria

Institute for Quantum Optics and Quantum Information, Austrian Academy of Sciences, A-6020 Innsbruck, Austria

Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria

QuTech and Kavli Institute of NanoScience, TU Delft, 2600 GA Delft, The Netherlands

Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Str. 1, 85748 Garching, Germany

## Appendix A Effective Spin-Spin Interactions

In this section, we analytically derive the expression for the effective coupling , as presented in the main text (MT).

General results.—We have introduced the effective spin-spin interaction as

(S1) |

with the spin-resonator coupling parameters given as . This yields

The sum over gives

where we have used the Fourier Series decomposition of the Dirac comb

(S2) | |||||

Given the range of integration over , only the first Dirac function contributes to . This leads to

(S3) | |||||

Note that in the standard situation ( being the spatial extent of the function ), the second term is negligible. Using the normalization property of , i.e., , we arrive at the result presented in the main text.

Box function.—For a box function , and assuming (and also the obvious condition ), the second term is exactly zero and we obtain , which does not depend on , nor the qubit positions.

## Appendix B Parametric Modulation of the Qubit-Resonator Coupling: Potential Advantages

In this Appendix we discuss the possibility to potentially boost and fine-tune the effective spin-spin interactions by parametrically modulating the longitudinal spin-resonator coupling.

Specifically, consider the generalization of Eq.(1) with an off-resonant modulation of at the drive frequency ,
i.e., , with
^{1}^{1}1If the driving amplitudes are zero for all but one specific mode, one recovers (approximately) a single-mode problem harvey18SM (); Royer2017SM ()..
When transforming to a suitable rotating frame and neglecting rapidly oscillating terms (in the limit )
we obtain a time-independent Hamiltonian which maps directly onto the system studied
so far with the replacements and .
Accordingly, for stroboscopic times synchronized with the detuning parameters (where with integer)
the unitary evolution in the lab frame reduces to
Eq.(4), up to a free evolution term
(which leaves the qubits untouched and even reduces to the identity as well if with integer),
with , the sign of which may be controlled by introducing relative phases between the driving terms harvey18SM (); Royer2017SM ().

Provided that parametric modulation of the qubit-resonator coupling (discussed as extension (iii) in the main text) can be implemented, it comes with the following potential advantages: (1) Here, the commensurability condition applies to the (tunable) detuning parameters rather than the bare spectrum . Therefore, even if the bare spectrum of the resonator is not commensurable, periodic disentanglement of the internal qubit degrees of freedom from the (hot) resonator modes can be achieved by choosing the driving frequencies appropriately. (2) The coupling can be amplified by cranking up the classical amplitudes , provided that for self-consistency. Moreover, is suppressed by the detuning only (rather than the frequencies as is the case in the static scenario). Still, the detuning should be sufficiently large in order to avoid photon-loss-induced dephasing Royer2017SM () and to keep the stroboscopic cycle time sufficiently short; see below for quantitative, implementation-specific estimates. (3) Since the number of modes effectively contributing to is well controlled by the choice , the high-energy cut-off problem described above is very well-controlled.

## Appendix C Timing-Induced Errors

In this Appendix we analyze errors induced by timing inaccuracies. Limited timing accuracy leads to deviations from the ideal stroboscopic times , with corresponding time jitter . For example, in quantum dot systems timing accuracies of a few picoseconds have been demonstrated experimentally bocquillon13 (). Here, we present analytical perturbative results that complement our numerical results as presented and discussed in the main text.

Our analysis starts out from the Hamiltonian given in Eq.(1) in the main text. For notational convenience we rewrite this Hamiltonian as

(S4) |

with . The time evolution operator generated by this Hamiltonian reads in full generality

(S5) | |||||

(S6) |

with the spin-dependent, multi-mode polaron transformation , as well as the single-qubit , and two-qubit gates , respectively. While for stroboscopic times , as discussed extensively in the main text, for non-stroboscopic times () generically will entangle the qubit and resonator degrees of freedom, with , thereby reducing the overall gate fidelity.

Errors due to limited timing accuracy will come from two sources: (i) First, as is the case for any unitary gate, there will be standard errors in the realization of single and two-qubit gates coming from limited timing control. For example, we can decompose the two-qubit gate as , where refers to the desired target gate and results in undesired contributions. The latter will be small provided that the random phase angles are small, i.e., . Accordingly, the timing control has to be fast on the time-scale set by the two-qubit interactions. A similar argument holds for the single qubit gate which is assumed to be controlled by spin-echo techniques. (ii) Second, for non-stroboscopic times there will be errors due to the breakdown of the commensurability condition (given by with integer); for non-stroboscopic times does not simplify to the identity matrix. This type of error is specific to our hot-gate scheme. While all errors of type (i) are fully included in our numerical calculations, within our analytical calculation presented here we will focus on errors of type (ii), as these are specific to our (quantum-bus based) hot gate approach.

In the following we will focus on errors due to the breakdown of the commensurability condition, as described by the unitary . Using the relation , we have

(S7) |

The qubits are assumed to be initialized in a pure state, . In the absence of errors, ideally they evolve into the pure target state defined as , which comprises both the single and two-qubit gates. As discussed above, here we neglect standard errors of type (i) and set at time , assuming that . Initially, the resonator modes are assumed to be in a thermal state, with , and . Then, the full evolution of the coupled spin-resonator system reads

(S8) | |||||

(S9) |

where refers to the qubit’s pure (target) density matrix at time in the case of ideal, noise-free evolution, while gives the density matrix of the coupled spin-resonator system in the presence of errors caused by incommensurate timing. The fidelity of our protocol is defined as

(S10) |

where denotes the trace over the resonator degrees of freedom. In order to derive a simple, analytical expression for the incommensurabiliy-induced error , in the following we restrict ourselves to a single mode, taken to be the mode (for small errors similar error terms due to multiple incommensurate modes can be added independently); also note that our complementary numerical results cover the multi-mode problem. Next, we perform a Taylor expansion of the undesired unitary as

(S11) |

with

(S12) |

This approximation is valid provided that the effective phase error is sufficiently small, that is ; approximately , where gives the thermal occupation of the mismatched mode. Then, up to second order in , we obtain

(S13) | |||||

where denotes the standard dissipator of Lindblad form. When tracing out the resonator degrees of freedom and computing the overlap with the ideal qubit’s target state , the first order terms are readily shown to vanish, and the leading order terms scale as (in agreement with our numerical results). Evaluating the second-order terms, we obtain a compact expression for the error given by

(S14) | |||||

Here, denotes the variance of the collective spin-operator in the ideal target state . Typically, for and the first term will dominate the overall error and we obtain

(S15) |

While the error scales linearly with the thermal occupation , it is suppressed quadratically for small phase errors and weak spin-resonator coupling . However, our analytical calculation is valid only provided that the Taylor expansion in Eq.(S11) is justified; again, this is the case if is satisfied. Still, our analytical treatment supports and complements our numerical results in the three following ways: (i) The timing error is quadratic in the time jitter , i.e., . (ii) The timing error is linearly proportional to the effective spin-spin interaction ; in agreement with our numerical results, (in the absence of dephasing) timing errors are suppressed for slow two-qubit gates. (iii) The timing error scales linearly with temperature .

## Appendix D Engineering of Spin Models

In this Appendix we provide further details regarding the implementation of targeted, engineered spin models.

Specifically, two more comments are in order: (i) For translation invariant models, the eigenstates of can be written as sine and cosine waves with normalized momentum . In particular for long-range models, we can obtain good approximations of using only a restricted number of cycles corresponding to the lowest spatial frequencies . (ii) To satisfy the condition , we can add to a diagonal component , which does not contribute to the dynamics, and which can also be used to improve the convergence with .

## Appendix E Additional Numerical Results

In this section, we present additional numerical results related to the realization of a phase gate between two distant qubits and the engineering of spin models (compare Figs. 2-3 of the main text).

Total photon number.—The total photon number in the transmission line is shown in Fig. S1(a), for the parameters of Fig. 2(a) of the main text. At short times, the qubits excite a number of photons ( for the chosen parameter set), which add up to the thermal background. These photons are then absorbed perfectly at the gate time . As shown in panel (b), the number of emitted photons tends to slightly decrease with increasing values of .

Fidelity—In panels (c-d) of Fig. S1 we provide further numerical results for spin-resonator coupling parameters , where the maximum fidelity is reached for later times (rather than at , as discussed in the main text), namely for and [panels (c) and (d), respectively]. In all cases considered we take the ratio such that a maximally entangling gate can (in principle) be achieved at . Taking , this is the case for , as required for a maximally entangling gate of the form . Since can only take on integer values, the value of needs to be fine-tuned in order to achieve a maximally entangling gate; without fine-tuning generically the target state will still be entangled (but not maximally entangled, even in the absence of noise). As shown in panels (c-d) of Fig. S1, periodic stroboscopic cycles for integer values of can clearly be identified. For values , many, small amplitude oscillations occur before the fidelity reaches its maximum value at the nominal gate time . In this parameter regime, the effective dynamics for typically feature a slow (secular), large amplitude with high-frequency, small amplitude oscillations on top; therefore, the relevant timescale for timing errors (due to timing inaccuracies ) is set by the interaction as , as exemplified in Fig. S1 (d) for . Since the essential dynamics appear on a long timescale , with only small changes occurring in the vicinity of , the constraints on timing errors are strongly relaxed, because stroboscopic precision on a timescale is not required in order to achieve a high-fidelity gate. Conversely, high-fidelity results can already be found in the parameter regime .

Nonlinear spectrum.—Next, we study potential errors due to a non-linear photonic spectrum (where ). Before presenting our detailed numerical results, some general comments are in order: (i) First, note that this type of error can only occur in the multi-mode setup, but is entirely absent in the single-mode regime, as could be (approximately) realized using parametric modulation of the qubit-resonator coupling harvey18SM (); Royer2017SM (). (ii) Second, the commensurability condition, as specified in the main text for a linear spectrum, can be generalized to spectra for which one can find a stroboscopic time (and integer multiples thereof), for which , etc. can be satisfied for integer values . This means that all fractions need to be rational numbers. Taking the ordering , we may summarize these conditions as . Then, with satisfied, all remaining equations can be deduced as . Therefore, given a specific spectrum , (in principle) one may still find specific (stroboscopic) times (and integer multiples thereof), for which the qubits disentangle entirely from the resonator modes, even if the spectrum is non-linear.

Our numerical results can be found in Fig. S1(e-f); here, we study the role of a nonlinear term in the dispersion relation of the transmission line, , where (for concreteness) we consider a quadratic term of the form . In panel (e), we represent the gate error versus time for and different values of (see legend). Around the gate time, the modes only partially synchronize, implying a minimal gate error which increases with . We further quantify these effects by representing in panel (f) the gate error (at such optimal time) as a function of , for the same values of . One clearly distinguishes two limits corresponding to (resp. ), which we can both understand analytically, considering for simplicity the effect of the asynchronicity of the mode ( is not affected by ), and . First, in the perturbative limit , the effect of the nonlinear term is analog to a timing error as discussed above, with the mode asynchronicity replacing the timing error in the expression of . This corresponds to a gate error

(S16) |

scaling thus as , as confirmed by our numerical simulations. In the opposite limit, , the mode asynchronicity hits a maximum value , and the error reads

(S17) |

scaling as , independently of , as also seen in our numerical simulations. This means that, along the lines of timing errors, the effect of nonlinear terms can be reduced by increasing .

Engineering of spin models.—In Fig. S2, we present additional numerical results on the engineering of spin models. In panel (a), we represent the formation of a spin glass with random interactions. In contrast to the models presented in Fig. 3 MT, one requires to implement the full spectrum, i.e., to use , to obtain a faithful generation of the target matrix. The convergence of the generated matrix with is shown in Fig. S2(b) for 1D models with nearest neighbor interactions and with power law decay . In both cases, we obtain a good representation of the targeted interactions for . Note that the convergence to NN interactions occurs at later times compared to the power-law case due to high spatial frequencies in the spectrum. As already shown in panel (a), to obtain a true spin glass model, one instead requires to implement the full spectrum of , see Fig. S2(c).

## Appendix F Decoherence Analysis

In this section, we provide detailed background material related to effects due to decoherence. First, we present the Master equation used in order to model decoherence in the form of qubit dephasing and resonator rethermalization. Next, we analytically derive an expression for the gate error caused by qubit dephasing. Thereafter, we numerically analyze rethermalization-induced errors. Finally, we show that the total error due to both (i) dephasing and (ii) rethermalization can be quantified in terms of a single cooperativity parameter.

### f.1 Master Equation

Master equation.—Within a standard Born-Markov approach, the noise processes described above can be accounted for by a master equation for the system’s density matrix as

(S20) | |||||

where describes the ideal (error-free), coherent evolution for longitudinal coupling between the qubits and the resonator mode, and is the pure dephasing rate. The second and third line describe rethermalization of the modes towards the a thermal state with an effective rate