Quantum Simulation and Optimization in Hot Quantum Networks

PHYSICAL REVIEW B 99, 241302(R) (2019)

Rapid Communications

Quantum simulation and optimization in hot quantum networks

M. J. A. Schuetz,1 B. Vermersch,2,3 G. Kirchmair,2,3 L. M. K. Vandersypen,4 J. I. Cirac,5 M. D. Lukin,1 and P. Zoller2,3 1Physics Department, Harvard University, Cambridge, Massachusetts 02318, USA 2Center for Quantum Physics, Institute for Experimental Physics, University of Innsbruck, A-6020 Innsbruck, Austria 3Institute for Quantum Optics and Quantum Information of the Austrian Academy of Sciences, A-6020 Innsbruck, Austria 4QuTech and Kavli Institute of NanoScience, TU Delft, 2600 GA Delft, The Netherlands 5Max-Planck-Institut für Quantenoptik, Hans-Kopfermann-Straße 1, 85748 Garching, Germany

(Received 13 September 2018; revised manuscript received 18 February 2019; published 27 June 2019)

We propose a setup based on (solid-state) qubits coupled to a common multimode 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 nonlocal quantum gates, the simulation of quantum spin models with engineered (long-range) interactions, and provides a ﬂexible architecture for the implementation of quantum approximate optimization algorithms.

DOI: 10.1103/PhysRevB.99.241302

Introduction. One of the leading approaches for scaling up of light. As opposed to transversal (Jaynes-Cummings-like) quantum information systems involves a modular architecture spin-resonator coupling [26,27], here we focus on longitu- that makes use of a combination of short- and long-distance dinal coupling as could be realized (for example) with su- interactions between the qubits [1,2]. In particular, long- perconducting qubits [8,28–31] or quantum-dot-based qubits distance interactions can be implemented via a quantum bus [8,9,32–36]. The Hamiltonian reads (¯h = 1) which can effectively distribute quantum information between N ω ∞ i z † z † remote qubits, as shown in the context of trapped ions [3–7], H = σ + ω a a + g , σ (a + a ), (1) lab 2 i n n n i n i n n solid-state systems [8,9], electromechanical resonators [10], i=1 n=1 i,n as well as circuit QED architectures [11–16]. In this Rapid σ Communication, we provide a unified theoretical framework with the qubit Pauli matrices i and gi,n the coupling strength for robust distribution of quantum information via a quantum between qubit i and mode n. We show below that for specific ∝ bus that operates at finite temperature [17], fully accounts for times t, which are integer multiples of the round-trip time t τ ≡ / the multimode structure of the data bus, and does not require 2L c, the dynamics of the qubits and all photons fully the qubits to be identical. Our approach [cf. Fig. 1(a)]re- decouple, while giving rise to an effective interaction between sults in an architecture where fully programmable interactions the qubits. between qubits can be realized in a fast and deterministic Analytical solution of time evolution. With the help of way, without any ground-state cooling requirements for the the spin-dependent, multimode displacement transformation † gi,n z † U = exp[ , σ (a − an )], in our model the spin dy- data bus, thereby setting the stage for various applications pol n i ωn i n in the context of quantum information processing [18]ina namics can be decoupled from the resonator dynamics (in the = † hot quantum network, different from quantum state transfer polaron frame), and we find Hlab UpolHpolUpol, where discussed previously [19–21]. As illustrated in Fig. 1(b), ω H = i σ z + ω a†a + J σ zσ z, (2) and discussed in detail below, one can use our scheme to pol 2 i n n n ij i j deterministically implement (hot) quantum gates between two i n i< j qubits. Moreover, we present a recipe to generate a targeted with the effective spin-spin interaction and scalable evolution for a large set of N qubits coupled g , g , J =− i n j n . via a single transmission line, thereby providing a natural ij 2 ω (3) architecture for the implementation of quantum algorithms, n n such as quantum annealing [22] or the quantum approximate Therefore, the evolution in the laboratory frame reads − − iHlabt = iHpolt † optimization algorithm (QAOA) [23–25], designed to find ap- e Upole Upol, as follows directly from a Tay- proximate solutions to hard, combinatorial search problems. lor expansion and U † U =1. Consider now the evolu- The model. We consider a set of qubits i = 1, 2,...,N pol pol tion at stroboscopic times tp = pτ (p positive integer), cor- with corresponding transition frequencies ω (typically in i responding to multiples of the round-trip time τ.Inthis the microwave regime) that are coupled to a (multimode) − ω † = case, all the modes synchronize,exp[itp n nanan] transmission line of length L; compare Fig. 1 for a schematic † † exp[−2πi npa an] = 1, since the number operators a an illustration. The transmission line is described in terms of pho- n n n feature an integer spectrum and ωntp = 2π pn; thus, the full tonic modes an with wave vectors kn = nπ/L, with a linear evolution reduces exactly to Ulab(tp) = exp[−iHlabtp], spectrum ωn = knc = nω1, where ω1 = πc/L is the frequency z z z −itp (ωi/2)σ −itp < Jijσ σ of the fundamental mode n = 1 and c is the (effective) speed Ulab(tp) = e i i e i j i j . (4)

(a) for |xi − x j| > a the effective interaction Eq. (3) simpliﬁes to 0 Jij = gig j/ω1 (cf. [40]). Accordingly, within this exemplary cos(knx) Transmission line model, the coupling Jij does not depend on a, nor the position −1 a of the qubits xi, and scales as L , showing that the time to qubits entangle qubits is only limited by the propagation time τ (∝L)

0 x1 x2 xN L x of light through the waveguide. (b)2 1 (c) −2 Applications. We now discuss three applications of our scheme, with a gradual increase in complexity, namely, (i) a Phase gate Max-Cut

C −3 remote qubits hot two-qubit phase gate, (ii) the engineering of spin models, H and (iii) the implementation of QAOA in the presence of VN entropy Concurrence 0 0 −4 decoherence and finite temperature. To this end, we consider 0.0 0.5 1.0 1.5 1 2 3 4 5 the possibility to potentially boost and fine-tune the effective t/τ M spin-spin interactions Jij by parametrically modulating the longitudinal spin-resonator coupling, as could be realized with FIG. 1. Hot quantum network. (a) Schematic illustration of N both superconducting qubits [8] or quantum-dot-based qubits qubits coupled to a transmission line of length L. (b) Dynamic evolu- [33](cf.[40] for further details). tion of two qubits, as exemplified for the von Neumann (VN) entropy Hot phase gate. As a first illustration, we consider the (left axis) and the concurrence (right axis) of the two-qubit density = = . = τ realization of a phase gate between two remote qubits N 2, matrix, with a 0 03L. At the round-trip time t , the qubits fully = = decouple from the waveguide and form a maximally entangled state, placed at each edge of the transmission line (x1 0, x2 − ρ =| |⊗ ρ even though the transmission line is far away from the ground state L a). Our initial state 0 0 0 n n consists√ of a pure initial qubit state with | =⊗ (|0+i |1) / 2 and (here, kBT = ω1). (c) Quantum approximate optimization algorithm 0 j j † ω (QAOA) with depth M solving Max-Cut with N = 6 qubits and ρ = − anan n − a thermal state of the waveguide with n exp( k T )[1 ω B a 4-regular graph (inset), and with decoherence (ideal case: blue; exp(− n )], and we use matrix-product-state (MPS) tech- γ / = . kBT dephasing with rate φ Jmax 0 003: orange; rethermalization with niques [42] to show numerically how the hot quantum network√ rate κ/| |=0.004: green), and at finite temperature kBT =ω1. generates the desired evolution, Eq. (4). We fix gi = ω1/ 8 which (under ideal circumstances) leads to a maximally entan- | = − π σ zσ z | Accordingly, for certain times the qubits fully disentangle gled pure state (t1) exp( i 4 1 2 ) 0 at the gate time = π/ = τ from the (thermally populated) resonator modes, thereby tg (4J12 ) after just one round-trip t1 (generalizations providing a qubit gate that is insensitive to the state of thereof are provided in [40]). In Fig. 1(b), we show the E C the resonator, while imposing no conditions on the qubit von-Neumann entropy and the concurrence of the two- ρ frequencies ω [37]. For specific times, the time evolution qubit density matrix 1,2, showing the realization of the i = in the polaron and the laboratory frame coincide and fully gate at t t1, in the presence of thermal occupation of the F decouple from the photon modes, allowing for the realization waveguide. The corresponding fidelity defined as overlap ρ | | of a thermally robust gate, without any need of cooling the of 1,2 with respect to the ideal state (t1) (t1) is shown transmission line to the vacuum [9]. Moreover, our approach in Fig. 2(a). In panels (b) and (c) both the mode occupation † † can be straightforwardly combined with standard spin-echo anan and the real-space occupation ax ax are displayed, < < techniques in order to cancel out efficiently low-frequency with ax,0 x L, referring to the discrete sine transform noise: By synchronizing fast global π rotations with the of an. In particular, the mode space picture [panel (b)] allows stroboscopic times tp, one can enhance the qubit’s coherence one to visualize the excitation of the linear spectrum of the waveguide, that synchronizes at time t = τ. Conversely, the times from the time-ensemble-averaged dephasing time T2 to the prolonged timescale T2. dynamics in real space [panel (c)] illustrates how qubit-qubit Frequency cutoff. In principle, the spin-spin coupling interactions are mediated by photon wave packets propagating = τ strength Jij as defined in Eq. (3)involvesall modes ballistically. At the round-trip time t , the waveguide n = 1, 2,..., naively leading to unphysical divergencies, as returns to its initial thermal state, as expected. In panel (d), discussed in the context of transversal qubit-resonator cou- we study the scaling of timing errors by showing the evolu- − F ≈ pling in Refs. [38,39]. In any physical implementation, tionoftheerror1√ around t tp. In the limit of small however, there is a microscopic lengthscale a that naturally errors | t| (a/c) ω1/J12, assuming a L, the numerical − F ≈ / 2 /ω 2 introduces a frequency cutoff. Specifically, we take the cou- results are well approximated by 1 4(c a) J12 1 t √ L = − (black line), with t = t − tp. Accordingly, the timing error pling parameters gi,n as gi,n gi n 0 cos(knx) f (x xi )dx, to account for the fact that the qubits couple to the local is sensitive to the cutoff a (as it controls the frequency scale voltage, where f (x − xi ) accounts for the microscopic spatial of the couplings), and scales linearly with the effective spin- extension of the qubit-transmission√ line coupling (cf. [40]for spin interaction J12, as slower dynamics are less vulnerable details); the factor ∼ n derives from the scaling of the rms to timing inaccuracies ∼ t; for further details, in particular zero-point voltage fluctuations with the mode index n, which related to the influence of temperature on timing errors, and −1 ω also implies gi ∝ L . In the examples below, we will con- effects due to nonlinear dispersion relations n,cf.[40]. = δ δ / sider for simplicity√ a box function f (x) x>0 x to gi,n = gi n(sin[kn(xi + a)] − sin[knxi])/(kna). Note that if cussion to the multiqubit case N 2 and provide a recipe the microscopic lengthscale a is set to zero, the summation how to generate a targeted and scalable unitary W = − σ zσ z over n in Eq. (3) does not converge. Instead for a finite a, and exp( i i< j wij i j ) with desired spin-spin interaction

241302-2 QUANTUM SIMULATION AND OPTIMIZATION IN HOT … PHYSICAL REVIEW B 99, 241302(R) (2019)

a† a 1.0 n n (a) (b) (a) (b)30 0 βm kBT/ω1 10 (N,d) (4, 3) 0.50 0 =(3, 2) m

0.8 20 −1 ,β 1 10 m 0.25 γ n F m 2 γ 0.6 10 10−2 (5, 2) 1 2 3 4 5 10−3 m 0.0 0.5 1.0 0.0 0.5 1.0 (c)Dephasing (d) Rethermalization t/τ t/τ 10−1 −1 † 10 1.0 a ax 0.10 (c) x (d) a/L 0 −2 −F 10 −F 10 0.1 −2 1 1 10 −1 0.2 10 0.05 10−3

0.5 −F 0.3 x/L 10−2 1 0.4 10−2 100 10−2 10−1 100 (γφN/Jmax)γMNd (κ[1+2nth(ω0)]/|Δ|)γMNd 10−3 0.00 0.0 0.5 1.0 −0.2 0.0 0.2 t/τ (c/a) J12/ω1Δt FIG. 4. Simulation of QAOA for Max-Cut, in the presence of decoherence. (a) d-regular graphs with N = 3, 4, 5 used for our FIG. 2. Hot phase gate between two distant qubits. (a),(b) numerical analysis of decoherence. Our graph with (N, d) = (6, 4) Fidelity as a function of time τ (a) for a = 0.03L and different trans- is shown in Fig. 1(c). (b) Optimization parameters γm,βm for N = 6, mission line temperatures 0 kBT 2ω1. (b) Mode and (c) real- M = 5. (c),(d) Scaling of errors with respect to the optimized QAOA space occupation as a function of the transmission line for a = 0.03L wave function |γ, β for (c) dephasing and (d) rethermalization. The and kBT = ω1, with ∼30 modes. (d) Error 1 − F around the gate dashed lines correspond to the scaling expressions given in the text. time tp for T = 0 and different values of the cutoff (legend) and For each panel, we consider the different graphs, depth M = 1, 3, 5, number of cycles p = 1, 4, 8, 16 (circles, crosses, stars, squares). Jmax/| |=0.02, 0.08. For (d), we consider kBT = 0,ω0. In (c) and For small timing errors, all data points collapse to a single curve (d), the dashed lines represent the curve y = x/2. 2 2 4(c/a) J12/ω1 t , shown as a black line. range spin model with power-law decay w = 1/|i − j|α parameters w . To this end, we consider a sequence ij ij (α = 1) and (b) a two-dimensional (2D) model with nearest- q = 1,...,η of successive cycles where for each strobo- neighbor (NN) interactions. The latter demonstrates that our scopic cycle (labeled by q) we may apply different coupling recipe allows for the realization of general spin models in amplitudes, i.e., g → g(q). For example, this could be done i i any spatial dimension and geometry (using a simple one- by parametrically modulating the spin-resonator coupling via dimensional physical setup). For both models, we observe microwave control [8,33]. The evolution at the end of the se- (η) z z the progressive emergence of the target spin interaction with quence is then given by Uη = exp(−it J σ σ ), with p i< j i, j i j increasing values for η, reaching the exact matrix at η = N. (η) = η (q) (q)/ω = η Ji, j q=1 gi g j 1, and trun tp being the total run The case of a spin glass with random interactions, and the con- time. A straightforward way to generate the desired unitary, vergence analysis with respect to η/N are presented in [40]. η i.e., to obtain w = J ( )t , consists in diagonalizing the target ij i, j p QAOA. Finally, we show how to generalize the techniques = N outlined above in order to implement quantum algorithms matrix as wij q=1 wqui,qu j,q in terms of real eigenvalues that provide approximate solutions for hard combinatorial wq and real eigenstates ui,q. This leads immediately to the con- (q) = ω / η = optimization problems such as Max-Cut (cf. Fig. 4 and [40]). dition gi wq 1 tpui,q to generate exactly W within N number of cycles, with t w /J , where J denotes As shown in Refs. [23,24], good approximate solutions to p q max max these kind of problems can be found by preparing the state the largest available spin-spin coupling [43]. In other words, |γ, β= β γ ··· β γ | β = we can engineer efﬁciently arbitrary spin-spin interactions Ux ( M )Uzz( M ) Ux ( 1)Uzz( 1) s , with Ux ( m ) −iβ σ x U γ = −iγ H H after a time t = Nt which only scales linearly with the exp[ m i i ], and zz( m ) exp[ m C], where C run p is the cost Hamiltonian encoding the optimization problem, number of qubits; trun = 2Ntp in the presence of spin echo. starting initially from a product of σ√x eigenstates, i.e., |s= These aspects are illustrated in Fig. 3, where we provide |−, −,... |− = | −| / examples for N = 25 and both (a) a one-dimensional long- , with ( 0 1 ) 2. In our scheme, this family of states can be prepared by alternating single-qubit operations Ux (βm ) with targeted spin-spin interactions gener- η η η wi,j → γ (a) =11 1.0 =25 (b)1.0 =25 1.0 ated as described above, with W Uzz( m ). Accordingly, for 20 200.8 200.8 0.8 QAOA we repeat our spin-engineering recipe M times with j 0.6 0.6 0.6 single-qubit rotations interspersed in between. This prepara- 10 100.4 100.4 0.4 tion step is then followed by a measurement in the com- 0.2 0.2 0.2 putational basis, giving a classical string z, with which one 0.0 0.0 0.0 10 i 20 10 20 10 20 can evaluate the objective function HC of the underlying combinatorial problem at hand. Repeating this procedure FIG. 3. Engineering of spin models. (a) Long-range interactions will provide an optimized string z, with the quality of the wij = 1/|i − j| and periodic boundary conditions, for η = 11, 25. result improving as the depth of the quantum circuit M is (b) 2D nearest-neighbor interactions with open boundary conditions, increased [23,24]. To illustrate and verify this approach, we with η = 25. Here, the indices i correspond to 2D indices i = (ix, iy ) have numerically simulated QAOA with up to N = 6 qubits of a square of 5 × 5 sites using the convention i = ix + 5iy. solving Max-Cut for several d-regular graphs with weights

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= (d) + δ Specifically, our scheme could be implemented based on wi, j wi, j d i,i, as depicted in Figs. 4(a) and 1(c), based on our model Hamiltonian given in Eq. (1), while accounting superconducting qubits or quantum-dot-based qubits coupled for both finite temperature and decoherence in the form of by a common high-quality transmission line, with details qubit dephasing and rethermalization of the resonator mode. given in [40]. For concreteness, let us consider quantum-dot- While our general multimode setup should (in principle) be based qubits [9,33–36] where longitudinal coupling could be well suited for the implementation of QAOA, here (in order modulated via both the detuning [33] or interdot tunneling to allow for an exact numerical treatment) we consider a sim- parameter [34], respectively. With projected two-qubit gate ∼ ≈ plified single-mode problem (with resonator frequency ω0), as times of 10 ns [33,34], a coherence time of T2 10 ms ω / π ≈ ∼ 6 could be realized using the resonance condition introduced by [46,47], and 0 2 1 GHz with quality factor Q 10 a monochromatically modulated coupling [8,33]. Specifically, [48–50], we estimate decoherence errors to be small ( 3%) = ω / σ z + † + for up to N ≈ 50 qubits and a QAOA circuit depth of M ≈ 10 we simulate the Hamiltonian H i( i 2) i a a g σ z ⊗ (a + a†) with controllable couplings g [8,33], for a graph with d ≈ 4, respectively, even in the presence i i i i ω ≈ detuning = ω0 − and Jij =−2gig j/ Jmax, supple- of nonzero thermal occupation withn ¯th( 0 ) 3. Further, mented by standard dissipators to account for (i) qubit dephas- the performance will be affected by timing errors, as is ing on a timescale ∼T2 = 1/γφ and (ii) rethermalization of the the case for any gate implementation. However, commercial resonator mode with an effective decay rate ∼κn¯th(ω0 )[44] equipment allows experiments with timing jitter of only a (cf. [40] for further details). As demonstrated in Fig. 1(c),for few picoseconds [40,51]. A similar analysis can be made for small-scale quantum systems (that are accessible to our exact superconducting qubits [40]. Note that these estimates might numerical treatment) our protocol efficiently solves Max-Cut be very conservative, as the essential figure of merit in QAOA with a circuit depth of M 5, finding the ground-state energy is not the quantum state fidelity F but the probability to find with very high accuracy (blue curve), corresponding to four the optimal (classical) bit-string z in a sample of projective { , ,...} cuts (shown in red in the inset), even in the presence of mod- measurements z1 z2 (obtained after many repetitions of erate noise [compare the cross and plus symbols in Fig. 1(c)]. the experiments). Decoherence and implementation. Based on our numerical Conclusion. To conclude, we have presented a protocol to findings and further analytical arguments, we now turn to the generate fast, coherent, long-distance coupling between solid- eventual limitations imposed by decoherence. Here, we focus state qubits, without any ground-state cooling requirements. on the QAOA protocol, since both our (i) hot gate (cf. [40]for While this approach has direct applications in terms of the en- a full decoherence-induced error analysis thereof) and (ii) the gineering of spin models (e.g., to implement QAOA) it would spin engineering protocol can be viewed as less demanding be interesting to further develop our theoretical treatment in limits of QAOA, where either M or N (or both) are small, order to increase the level of robustness of our scheme, e.g., to thereby yielding comparatively smaller errors because of a apply protocols based on error correcting photonic codes [54], shorter run time; for example, for the two-qubit phase gate which can protect against photon losses or rethermalization. = = Yet another interesting research direction would be to adapt M 1, N 2. The total QAOA run time trun can be upper- ≈ γ / γ = / γ our scheme to other physical setups, say solid-state defect bounded as trun MNd Jmax, with 1 M m m and the factor Nd/Jmax corresponding to the (maximum) time centers coupled by phonons [10]. required to implement all eigenvalues wq d of the Max-Cut Acknowledgments. We thank S. Harvey, H. Pichler, P. problem. To keep decoherence effects minimal, this timescale Scarlino, D. Vasilyev, S. Wang, and L. Zhou for fruitful should be shorter than all relevant noise processes. The ac- discussions. Numerical simulations were performed using cumulated dephasing-induced error can be estimated as ξφ ∼ the ITensor library [55] and QuTiP [56]. M.J.A.S. would γφN × γ MNd/Jmax, where ∼γφN is the effective many-body like to thank the Humboldt Foundation for financial support. dephasing rate (cf. [40]); as shown in Fig. 4(c),wehave L.M.K.V. acknowledges support by an ERC Synergy grant numerically confirmed this scaling for all graphs shown in (QC-Lab). J.I.C. acknowledges the ERC Advanced Grant Figs. 4(a) and 1(c). Similarly, as demonstrated in Fig. 4(d), QENOCOBA under the EU Horizon 2020 program (Grant the indirect rethermalization-induced dephasing error, me- Agreement No. 742102). Work in Innsbruck is supported by diated by incoherent evolution of the resonator mode, can the ERC Synergy Grant UQUAM, the SFB FoQuS (FWF be quantified as ξκ ∼ κeff × γ MNd/| |, with total linewidth Project No. F4016-N23), and the Army Research Laboratory κeff = κ[2¯nth(ω0 ) + 1]. The total decoherence-induced error Center for Distributed Quantum Information via the project ξ = ξφ + ξκ can then be optimized with√ respect to , yielding SciNet. Work at Harvard University was supported by NSF, the compact expression ξ ≈ γ dMN3/2/ C, with the cooper- Center for Ultracold Atoms, CIQM, Vannevar Bush Fel- 2 lowship, AFOSR MURI and Max Planck Harvard Research ativity C = g /(γφκeff ). With this expression, we can bound the maximum number of qubits N and circuit depth M for a Center for Quantum Optics. given physical setup with cooperativity C. M.J.A.S. and B.V. contributed equally to this work.

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of ∼100 MHz the thermal occupation amounts to ∼20 thermal eigenvalue wq = π/4, such that tp = tcoh = π/4Jmax. photons, even at very cold dilution fridge temperatures of [44] We ignore single spin-relaxation processes, since the associated

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[51] Consider (for example) fundamental frequencies of ∼100 MHz, [53] E. Bocquillon, V. Freulon, J. M. Berroir, P. Degiovanni, B. as realized experimentally in Ref. [41], which translates to Plaçais, A. Cavanna, Y. Jin, and G. Fève, Science 339, 1054 round-trip times of ∼10 ns. This timescale is much longer than (2013). state-of-the-art timing accuracies t of a few picoseconds as [54] M. H. Michael, M. Silveri, R. T. Brierley, V. V Albert, J. demonstrated experimentally in Refs. [52,53], leaving us with Salmilehto, L. Jiang, and S. M. Girvin, Phys.Rev.X6, 031006 −4 very small relative time jitter (ω1/2π ) t ∼ 10 . (2016). [52] O. E. Dial, M. D. Shulman, S. P. Harvey, H. Bluhm, V. [55] See http://itensor.org. Umansky, and A. Yacoby, Phys.Rev.Lett.110, 146804 [56] J. R. Johansson, P. D. Nation, and F. Nori, Comput. Phys. (2013). Commun. 184, 1234 (2013).

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