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PHYSICAL REVIEW X 10, 021060 (2020) Featured in Physics

Efficient Multiphoton Sampling of Molecular Vibronic Spectra on a Superconducting Bosonic Processor

Christopher S. Wang,1,2,* Jacob C. Curtis,1,2 Brian J. Lester ,1,2 Yaxing Zhang,1,2 Yvonne Y. Gao,1,2 Jessica Freeze ,3 Victor S. Batista,3 Patrick H. Vaccaro,3 Isaac L. Chuang,4 Luigi Frunzio ,1,2 ‡ † Liang Jiang,1,2, S. M. Girvin ,1,2 and Robert J. Schoelkopf1,2, 1Departments of Physics and Applied Physics, Yale University, New Haven, Connecticut 06511, USA 2Yale Quantum Institute, Yale University, New Haven, Connecticut 06520, USA 3Department of Chemistry, Yale University, New Haven, Connecticut 06511, USA 4Department of Physics, Center for Ultracold Atoms, and Research Laboratory of Electronics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

(Received 16 December 2019; revised manuscript received 30 March 2020; accepted 27 April 2020; published 17 June 2020)

The efficient simulation of quantum systems is a primary motivating factor for developing controllable quantum machines. For addressing systems with underlying bosonic structure, it is advantageous to utilize a naturally bosonic platform. Optical passing through linear networks may be configured to perform quantum simulation tasks, but the efficient preparation and detection of multiphoton quantum states of light in linear optical systems are challenging. Here, we experimentally implement a boson sampling protocol for simulating molecular vibronic spectra [J. Huh et al., Nat. Photonics 9, 615 (2015)]in a two-mode superconducting device. In addition to enacting the requisite set of Gaussian operations across both modes, we fulfill the scalability requirement by demonstrating, for the first time in any platform, a high-fidelity single-shot number resolving detection scheme capable of resolving up to 15 photons per mode. Furthermore, we exercise the capability of synthesizing non-Gaussian input states to simulate spectra of molecular ensembles in vibrational excited states. We show the reprogrammability of our implementation by extracting the spectra of photoelectron processes in H2O, O3,NO2, and SO2. The capabilities highlighted in this work establish the superconducting architecture as a promising platform for bosonic simulations, and by combining them with tools such as Kerr interactions and engineered dissipation, enable the simulation of a wider class of bosonic systems.

DOI: 10.1103/PhysRevX.10.021060 Subject Areas: Quantum Physics,

I. INTRODUCTION directly exploited. A prominent example of this is the manipulation of bosonic atoms in an to Simulation of quantum systems with quantum hardware – offers a promising route toward understanding the complex explore various types of many-body physics [2 4]. properties of those systems that lie beyond the computa- Boson sampling is another example of a computationally tional power of classical computers [1]. A particularly challenging task that can be performed by manipulating efficient approach is one that utilizes the natural properties bosonic excitations [5,6]. Conventionally described in the of the quantum hardware to simulate physical systems that context of linear optics, boson sampling, in its many forms, share those properties. This approach serves as the foun- involves the single photon detection of nonclassical states dation for bosonic quantum simulation, where the natural of light passing through a linear interferometric network. statistics and interference between bosonic excitations are Current technologies for optical waveguides allow the creation of complex interferometers across many modes. An outstanding challenge in the optical domain, however, is generating and detecting nonclassical states of light with *[email protected] † high efficiencies. [email protected] ‡ Current address: Pritzker School of Molecular Engineering, In the microwave domain, nonlinearities provided by University of Chicago, Chicago, Illinois 60637, USA. Josephson junctions enable the powerful preparation and flexible quantum nondemolition (QND) measurement of Published by the American Physical Society under the terms of quantum states of light in superconducting circuits [7,8]. the Creative Commons Attribution 4.0 International license. The ability to perform QND photon number measurements Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, not only enables the high-fidelity measurement of bosonic and DOI. via repeated detections [9,10], but also enables the

2160-3308=20=10(2)=021060(18) 021060-1 Published by the American Physical Society CHRISTOPHER S. WANG et al. PHYS. REV. X 10, 021060 (2020) construction of more complex measurement operators. a distinct manifold of vibrational eigenstates. The normal The circuit QED platform has also demonstrated universal modes of vibration for a given electronic state are obtained control over individual bosonic modes as well as robust by expanding the PES in powers of displacement coor- beam splitter operations between separate modes [11–13]. dinates referenced to the minimum-energy (equilibrium) These capabilities motivate an alternative approach to configuration and retaining only up to quadratic terms. performing both boson sampling [14] and bosonic quantum Under this harmonic approximation, the corresponding simulation protocols [15] using superconducting circuits. transformation of the set of creation and annihilation ˆ An instance of bosonic quantum simulation that maps operators a ¼ðaˆ 1; …; aˆ NÞ for N vibrational modes may onto a generalized boson sampling problem is obtaining be expressed using the Duschinsky transformation [22] molecular vibronic spectra associated with electronic tran- which can be decomposed into Gaussian operations via the sitions [16]. The requires, in addition to beam Doktorov operator [23]: splitters, single-mode displacement and squeezing opera- aˆ → ˆ aˆ ˆ † tions. Furthermore, the output will generally be a multi- UDok UDok; ð1Þ mode multiphoton state, thus requiring a series of photon ˆ Dˆ α Sˆ † ζ0 Rˆ Sˆ ζ number resolving detectors at the output. Experimental UDok ¼ ð Þ ð Þ ðUÞ ð Þ; ð2Þ imperfections in the controls, photon loss, and detector inefficiency have made a linear optical implementation of where this algorithm challenging [17]. Recent work leveraging the ˆ ˆ ˆ ˆ bosonic nature of two modes of a single trapped ion DðαÞ¼Dðα1Þ ⊗ Dðα2Þ ⊗ ⊗ DðαNÞ; ð3Þ has been successful in generating more accurate spectra 0 0 0 [18]; however, an efficient detection scheme capable of ˆ ð†Þ ð0Þ ˆ ð†Þ ð Þ ˆ ð†Þ ð Þ ˆ ð†Þ ð Þ S ðζ Þ¼S ðζ1 Þ ⊗ S ðζ2 Þ ⊗ ⊗ S ðζ Þð4Þ directly sampling from the exponentially growing bosonic N Hilbert space remains elusive. Alternative approaches to correspond to a tensor product of single-mode displace- this problem that directly emulate an electronic transition ment and squeezing operations across all N modes, in a superconducting architecture via quenching dynamics respectively. Rˆ ðUÞ is an N-mode rotation operator corre- have been proposed [19], but such quenching may be sponding to a N × N rotation matrix U which can be experimentally challenging to implement with high pre- decomposed into a product of two-mode beam splitter cision on a large scale. operations (see Appendix A) [24].ForN ¼ 2, U is a two- Previous experimental work on simulating molecular dimensional rotation matrix parametrized by a single vibronic spectra in superconducting systems has been angle θ. The set of dimensionless Doktorov parameters restricted to a single vibrational mode [20,21]. Here, we 0 0 0 α ¼ðα1; …; α Þ, ζ ¼ðζ1; …; ζ Þ, ζ ¼ðζ ; …; ζ Þ and U experimentally implement a two-mode superconducting N N 1 N originate from molecular structural information in the bosonic processor with a full set of controls that enables different electronic configurations (see Appendix A). the scalable simulation of molecular vibronic spectra. The ˆ ψ processor combines arbitrary (Gaussian and non-Gaussian) Applying UDok to an initial state j 0i of the bosonic state preparation and a universal set of Gaussian operations processor directly emulates the physical process of a enabled by four-wave mixing of a Josephson potential. pretransition molecular vibrational state experiencing a Most importantly, we implement a single-shot QND photon sudden change in the electronic PES and being expressed number resolving detection scheme capable of resolving in the post-transition vibrational basis. This projection gives rise to the Franck-Condon factors (FCFs) defined the first n ¼ 16 Fock states per mode. This detection max as the vibrational overlap integrals of an initial pretransition scheme, when operated without errors on a multiphoton ⃗ … distribution bounded by n states per mode, extracts the vibrational eigenstate, jni¼jn; m; i, with a final post- max ⃗0 0 0 … maximum possible amount of information from the under- transition vibrational eigenstate, jn i¼jn ;m; i: lying spectra per run of the experiment. ⃗0 ˆ ⃗ 2 FCFjn;⃗n⃗0 ¼jhn jUDokjnij : ð5Þ II. BOSONIC ALGORITHM FOR In spectroscopic experiments, the validity of this “sudden FRANCK-CONDON FACTORS approximation” depends on the vastly different energy The mapping of molecular vibronic spectra onto a scales that typically characterize electronic and nuclear bosonic simulation framework can be understood by motions (≫10 000 cm−1 versus ∼100–1000 cm−1, respec- considering the nature of vibrational dynamics that accom- tively). By further assuming that the transition electric panies an electronic transition. In keeping with the tenets of dipole moment does not depend on nuclear coordinates the adiabatic Born-Oppenheimer approximation, the pre- (i.e., the Condon approximation), the relative intensities of sumed separability between electronic and nuclear degrees features appearing in vibrationally resolved absorption of freedom results in distinct electronic states, each of and emission spectra will be directly proportional to the which forms a potential-energy surface (PES) that supports corresponding FCFs. Indeed, the practical importance of

021060-2 EFFICIENT MULTIPHOTON SAMPLING OF MOLECULAR … PHYS. REV. X 10, 021060 (2020) these quantities often stems from the structural and Hamiltonian. In our case where N ¼ 2, the rotation dynamical insights that they can provide about excited operator corresponds directly to enacting a single beam electronic states, information that can be challenging to splitter. Both the single-mode squeezing and beam splitter obtain through other conventional spectroscopic means. operations utilize the four-wave mixing capabilities of the coupler . Two pump tones that fulfill the appro- ω ω 2ω III. EXPERIMENTAL IMPLEMENTATION priate frequency matching condition ( 1 þ 2 ¼ A=B for the squeezing operation and ω2 − ω1 ¼ ωB − ωA for the Our superconducting processor is designed to manipu- ω ˆ beam splitter operation, where 1=2 are the two pump late the bosonic modes of two microwave cavities, cA and ω ˆ frequencies and A=B are the cavity frequencies) are sent cˆB (Fig. 1). In our design, a coupler transmon tC dis- persively couples to both cavities, enabling beam splitter through a port that primarily couples to the coupler trans- [13] and squeezing operations through driven four-wave mon, which enacts the desired Hamiltonians [13]: mixing processes. The coupler transmon is also disper- ˆ ℏ iφ ˆ ˆ† −iφ ˆ† ˆ HBS= ¼ gBSðtÞðe cAc þ e c cBÞ; ð6Þ sively coupled to a readout resonator rˆC. Displacement B A operations on the cavity modes are performed via resonant ϕ 2 − ϕ †2 ˆ ℏ i i ˆ i i ˆ drives through local coupling ports. Additional ancillary Hsq;i= ¼ gsq;iðtÞðe ci þ e ci Þ; ð7Þ ˆ ˆ transmon-readout systems ftA; rˆAg and ftB; rˆBg are inserted into the device and couple to each cavity for the where i ∈ fA; Bg. Importantly, the phase of the operations φ ϕ ϕ purposes of state preparation and tomography. f ; A; Bg implemented by these Hamiltonians is con- The goal of the present simulation is to emulate the trolled by the phase of the pump tones, allowing the transformation of a molecular vibrational state due to an microwave control system to generate the correct family electronic transition using the photonic state of the quantum of beam splitter and squeezing operations for performing ˆ processor. The processor is first initialized in a state jψ 0i UDok. A set of transmon measurements is then carried out corresponding to the pretransition molecular vibrational for the purpose of postselecting the final data on measuring state of interest. A vacuum state of both bosonic modes is all in their ground states. This verification step prepared through feedback cooling protocols, and Fock primarily aims to reject heating events of the transmons out states jn; mi are initialized using optimal control techniques of their ground state; the heating of these ancillas otherwise [12] (see Appendix D). The ability to reliably synthesize dephases the cavities while coupler heating effectively halts arbitrary Fock states translates to the powerful capability of the pumped operations by shifting the requisite frequency simulating FCFs starting from vibrationally excited states, a matching conditions. This postselection also serves to task which is challenging in most other bosonic simulators. ensure that the ancillas begin in their ground states for The Doktorov transformation is then applied, producing a the subsequent measurement of the cavities. In our experi- basis change to that of the post-transition vibrational ment, we reject 5%–10% of the data depending on which

FIG. 1. Circuit schematic of the superconducting bosonic processor. The device consists of two microwave cavity modes (blue cˆA, and green cˆB) which represent the symmetric-stretching and bending modes of a triatomic molecule in the C2v point group, respectively. ˆ ˆ Ancilla measurement and control modules (shaded in blue and green) consisting of a transmon ðtA; tBÞ and readout resonator ˆ ˆ ðrA; rBÞ couple to each cavity mode for state preparation and measurement. The coupler module (shaded in red) consists of a coupler ˆ transmon tC and readout resonator rˆC. The coupler transmon is used to facilitate bilinear Gaussian operations on the two cavity modes through four-wave mixing, and the readout resonator is used for both characterization and postselection. This configuration is extensible to a general linear array of N cavity modes with nearest-neighbor coupler modules and N ancillary modules, as depicted in light gray.

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TABLE I. Full set of controls of the bosonic processor. The subscripts i ∈ fA; Bg correspond to operations on each cavity and their respective ancillary modules. κA and κB are the intrinsic linewidths of the cavity modes. Optimal control pulses that utilize both resonant cavity and ancilla drives with complex envelopes, εiðtÞ and ϵiðtÞ, respectively, generate a desired initial (see Appendix D). 0 0 jn ;mi denotes a particular joint photon number to be measured in the cavities, and fbig represent the bits associated with the binary decomposition of the photon number. For example, j5i¼j0101i. The third column presents an estimate of the expected limits on fidelity due to photon loss for the hardware in this current implementation of the simulator.

Interaction Error rate State preparation Optimal control ˆ ε ˆ ε ˆ† ϵ ˆ ϵ ˆ† κ τ ∼ 10−3 − 10−2 HdriveðtÞ¼ i ðtÞci þ iðtÞci þ i ðtÞti þ iðtÞti i prep;i Operations Displacement ε˜ ˆ ε˜ ˆ† κ τ ∼ 10−4 i ðtÞci þ iðtÞci i disp;i − ϕ 2 ϕ †2 −2 Squeezing i i ˆ i i ˆ κ ∼ 5 10 gsq;iðtÞðe ci þ e ci Þ i=gsq;i × Beam splitter iφ ˆ ˆ† −iφ ˆ† ˆ κ τ ∼ 10−2 gBSðtÞðe cAcB þ e cAcBÞ i BS Measurement ˆ 0 0 0 0 ˆ ˆ ˆ κ τ ∼ 10−3 − 10−2 Single-bit extraction M0 ¼jn ;mihn ;mj, M1 ¼ 1 − M0 i meas 0 κ τ ∼ 10−2 − 10−1 Sampling jn i¼jb3;b2;b1;b0i i meas transformation is simulated. Finally, averaging many mea- in the experiment, which give selectivities above 99.9% and surements of the cavities in a photon number basis, implement the followingP mapping for a general state of the 0 0 ψ fn ;mg, gives the desired FCFs. The full set of controls two cavities j i¼ i;j cijji; ji for a chosen set of photon is detailed in Table I; the relatively small error rates due to numbers fn0;m0g to probe: photon loss provide a sense of scale for attainable circuit X depths while maintaining high-fidelity performance (see cijji;ji ⊗ jg;gi Sec. VI C). i;j X X 0 → c i;j ⊗ g;g c 0 i;m ⊗ g;e IV. MEASUREMENT PROTOCOLS ijj i j iþ im j i j i ≠ 0 ≠ 0 ≠ 0 i n ;jXm i n Two complementary measurement schemes are used to 0 0 0 þ c 0 jn ;ji ⊗ je;giþc 0 0 jn ;m i ⊗ je;ei; ð8Þ extract FCFs from the final state of the processor, both of n j n m j≠m0 which utilize the dispersive coupling of each microwave cavity to itsP ancillary transmon-readout system [Fig. 2(a)]: where jgi and jei are the ground and first excited state of ˆ ℏ − χ ˆ† ˆ ˆ†ˆ Hint= ¼ i∈fA;Bg ici citi ti, where the dispersive inter- each ancilla transmon. The transmons are then individually action strengths in our experiment are χA ¼ 2π × 748 kHz read out using standard dispersive techniques, and the χ 2π 1240 and B ¼ × kHz. Fundamentally, the difference results are correlated on a shot-by-shot basis to extract a between these two measurement schemes arises from the single bit of information for each joint photon number state ability of the latter to extract more than one bit of probed. We thus call this measurement scheme single-bit information on a given single shot of the experiment. extraction. Extracting FCFs using the single-bit extraction scheme, A. Single-bit extraction however, is not scalable. The bosonic Hilbert space grows N The first scheme [Fig. 2(b)] maps a given joint cavity exponentially with the number of modes N as nmax, where photon number population fn0;m0g onto the joint state of nmax is the maximum number of Fock states considered for 16 the two transmons via state-selective π pulses. These pulses each mode (so for instance, nmax ¼ corresponds to ω ω0 − 0χ 02 − 0 χ0 2 j0i − j15i). Thus, any measurement protocol that only have frequencies tA ¼ tA n A þðn n Þð A= Þ and ω ¼ ω0 − m0χ þðm02 − m0Þðχ0 =2Þ, where the small extracts a single bit of information from the underlying tB tB B B distribution at a time must necessarily query the exponen- second-orderP dispersive shift is also taken into account: ˆ 0 0 † † ˆ†ˆ tially growing number of final states. This is the case for a H =ℏ ¼ ∈ ðχ =2Þcˆ cˆ cˆ cˆ t t . In our experiment, int i fA;Bg i i i i i i i recent implementation of this simulation using two phonon χ0 2π 1 31 χ0 2π 1 35 A ¼ × . kHz and B ¼ × . kHz. The pulses modes of a single trapped ion [18]. are applied with a Gaussian envelope truncated at 2σt such that the bandwidth is approximately σf ¼ 1=ð2πσtÞ, B. Photon number resolved sampling where σt is the standard deviation of the pulse in time. The selectivity is defined as the probability of exciting the In optical systems, detectors that can simultaneously ancilla given occupation in an adjacent photon number state resolve the photon number and do so with a high quantum of interest. Pulses with σt ¼ 1 μs for both ancillas are used efficiency [25,26] are a necessary ingredient for protocols

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(a) (b)

Single-bit extraction

nn

(c)

b b1 b2 b3 0 1 01 01 01 Sampling

67 8 14 0 15 n P

n 02468 20614141 6 6

FIG. 2. QND measurements of cavity photon number. (a) Depiction of two 3D λ=4 coaxial cavities dispersively coupled to individual ancillary modules (outlined in black), each consisting of an ancilla qubit and a readout resonator. From the point of view of the measurement, the ancillary modules serve as reconfigurable black boxes used to detect the photon number in each cavity. The dotted trace serves to illustrate that the state of the two cavities can generally be entangled. (b) Single-bit extraction. In this scheme, on a given run of the experiment, each ancilla is excited conditioned on a predetermined photon number n in its respective cavity. (c) Sampling. Here, instead of having a binary output, each detection module serves as a photon number resolving detector. Sequential QND measurements of the operators associated with the first four bits of each photon number’s binary decomposition resolve up to 15 photons per mode. For a general state of the cavity, each measurement projects the state into the eigenspace of the outcome, ultimately projecting out a single Fock state with its corresponding probability. In the schematic, a sequence sampling the 6 photon component (j6i¼j0110i) from a displaced Fock state is shown. The state is first probabilistically measured to be in the b0 ¼ 0 (even parity) subspace, thus projecting out only even photon numbers. The next measurement further projects this state into the b1 ¼ 1 subspace, and so on, until the measurement converges on a single photon number.

ˆ such as [27] and communication based operators Pk in the cavity Hilbert space whose eigenvalues λ 1 0 1 on repeater networks [28], among others. The boson k; ¼ correspond to bk ¼f ; g, respectively, with the sampling algorithm implemented in this work also requires following matrix elements ij: photon number resolved detection, as the output distribu- 8 > tion usually has multiple photons per mode. <> 0 if i ≠ j Our second measurement scheme implements single- ðPˆ Þ ¼ i ð11Þ shot photon number resolving detection [Fig. 2(c)] that k ij > 1 − 2 ðmod 2Þ if i ¼ j; : 2k overcomes the scalability issue associated with the single- bit extraction method. At the heart of this technique is the dispersive Hamiltonian between each cavity mode cˆ and where b·c denotes the floor function. We synthesize a set of ϵ ∈ 0 1 2 3 their respective transmon ancilla tˆ: optimal control pulses kðtÞ, k f ; ; ; g, that excite the transmon from its ground state jgi to the excited state jei H=ˆ ℏ ¼ −χcˆ†cˆtˆ†t:ˆ ð9Þ conditioned on the cavity state’s projection onto the parity ˆ operators Pk. In our experiment, these pulses are numerically 16 It can be shown that driving the transmon with numeri- optimized for the Hilbert space of each cavity up to nmax ¼ cally optimized waveforms ϵðtÞ, and have a duration between 800 and 1200 ns. The QND nature of each of these mapping pulses on the ˆ ℏ ϵ ˆ ϵ ˆ† Hdrive= ¼ ðtÞt þ ðtÞt ; ð10Þ cavity state ensures that the pulses can be applied sequen- tially (with transmon measurements followingP each pulse) enables the QND mapping of any binary valued operator of ψ 15 to project an initial cavity state j i¼ n¼0 cnjni with the cavity Hilbert space onto the state of the transmon [11,12]. 16 bounded support within nmax ¼ into a definite Fock Our strategy for measuring the number of photons for a 2 state jni with a probability jcnj . In order to minimize errors given cavity state is to representP photon number in binary due to decoherence when the transmon is excited, after the k k Π max max 2bk ∈ 0 1 jni¼j k¼0bki,wheren ¼ k¼0 and fbkg f ; g, kth bit is mapped onto the transmon and measured, the and then to sequentially measure each bit bk on a given run of transmon is reset to its ground state using real-time feed- the experiment. This amounts to identifying a set of parity forward control to prepare for the measurement of the

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(k 1)th bit of the same cavity state. This protocol is (a) þ Ideal spectra performed simultaneously on each cavity, which returns a Single-bit sample from the underlying joint photon number distribu- Sampling tion on a shot-by-shot basis. This measurement scheme O thus implements bosonic sampling of the final state H H distribution, which we simply call sampling. These binary detectors optimally resolve the photon number in the cavities in N log2ðnmaxÞ measurements, but are prone to more errors due to transmon decoherence that are, in Relative intensity principle, correlated bit to bit. Errors in the measurement of individual bits can lead to unintuitive misassignments of the photon number—for instance, j8 ¼ 1000i would be Wave number (cm–1) misassigned as j0 ¼ 0000i if a single error misassigns b3. We leave the task of characterizing these errors in full (b) detail, optimizing the technique, and potentially applying deconvolution methods [29] to improve the accuracy of the sampled distribution as the subject of future work. O The advantage of the sampling protocol compared to the OO single-bit extraction method can thus be seen by comparing the required number of simulator runs for a desired statistical error on all output probabilities (see Supplemental Material

N Relative intensity [30]). Specifically, the latter method requires a factor of nmax more runs, directly reflecting the need to independently query each state in the entire Hilbert space. Wave number (cm–1) V. SIMULATED PHOTOELECTRON SPECTRA FIG. 3. Experimental Franck-Condon factors. Measured data for The full set of FCFs for a given electronic transition can ˜ 2 (a) photoionization of water to the ðB B2Þ excited state of the provide, within the Condon approximation, the relative ν h þ ˜ 2 − intensities of vibronic progressions appearing in corre- cation H2O → H2O ðB B2Þþe starting in the vacuum (vibra- sponding photoelectron spectra. The algorithm exploited in tionless, n ¼ 0, m ¼ 0) state and (b) the photodetachment of this work extracts FCFs under the commonly deployed the ozone anion to the ground state of the neutral species − hν − harmonic approximation. As such, the resulting relative O3 → O3 þ e starting from a vibrational eigenstate possessing intensities will deviate from those measured in actual one quantum of symmetric-stretching and two quanta of bending molecular spectra owing to the effects of vibrational excitation (n ¼ 1, m ¼ 2). The abscissa scale corresponds to anharmonicity in the true PES. We simulate four different vibrational term values (energies in cm−1) within the final (post- photoelectron processes starting in various vibrational transition) electronic PES calculated from the harmonic frequen- ν cies for symmetric-stretching and bending degrees of freedom: ψ →h þ ˜ 2 − initial states j 0i¼jn; mi:H2O H2O ðB B2Þþe , ν˜ ¼ n0ν˜0 þ m0ν˜0 . Solid lines depict theoretical FCFs, arti- ν ν ν stretch bend − h − − h − h þ − 10 −1 O3 → O3 þ e ,NO2 → NO2 þ e , and SO2 → SO2 þ e . ficially broadened with Lorentzian profiles ( cm FWHM). Experimental results for photoionization of water starting Circles represent experimental data using the single-bit extraction (purple) and sampling (red) measurement schemes explained in the in jψ 0i¼j0; 0i and photodetachment of the ozone anion main text; statistical error bars for the latter measurement are not starting in jψ 0i¼j1; 2i, respectively, are presented in visible on this scale (see Supplemental Material [30]). Systematic Fig. 3 (see Supplemental Material for additional data [30]). errors associated with transmon decoherence during the selective π We highlight the results for these two processes due to the pulses are corrected for (see Appendix E). Additional errors are long vibronic progressions they possess, thus verifying the present in the sampled values, owing to decoherence effects during capability of our simulator to generate and measure large the binary decomposition measurement chain. photon numbers. The particular electronic state of the water ˜ 2 cation considered here ðB B2Þ is the second excited state of doublet multiplicity, with the attendant electronic restricted to the two-dimensional subspace of the symmet- wave function having B2 symmetry. The other processes consider transitions beginning and ending in electronic ric-stretching and bending modes. ground states. All of the triatomic molecules targeted here A figure of merit for quantifying the quality of the quantum simulation is the distance D ¼ retain C2v point-group symmetry for their equilibrium P P 1 nmax nmax meas − ideal configurations in both the pretransition and post-transition 2 i¼0 j¼0 jpij pij j between the measured prob- meas ideal electronic states, thereby ensuring that analyses can be abilities fpij g and the ideal distribution fpij g. The

021060-6 EFFICIENT MULTIPHOTON SAMPLING OF MOLECULAR … PHYS. REV. X 10, 021060 (2020) distances for the two simulated processes in Fig. 3 are D ¼ Therefore, the formal task of obtaining the full 0.049 (H2O) and 0.105 (O3) for the single-bit extraction Franck-Condon profile, which by itself contains an expo- scheme and D ¼ 0.152 (H2O) and 0.148 (O3) for the nential number of entries corresponding to the size of the N sampling scheme. The sampling distance will ultimately be Hilbert space nmax, can be seen to be at least exponentially the relevant figure of merit for evaluating the practical challenging. performance of this approach as it is scaled up. The other In practice, however, electronic spectra of many polya- relevant metric is run time. As previously discussed in tomic molecules initially cooled to near their rovibronic Sec. IV B, the single-bit extraction scheme requires a factor ground states typically exhibit vibronic progressions con- N of nmax more runs compared to the sampling scheme for a fined to a small region in the vibrational Hilbert space. This desired statistical error. Our experiment operates at an stems primarily from two effects. First, a single electronic effective repetition rate of roughly ∼300 Hz, thus requiring transition in large molecules typically does not induce large ∼7 h versus ∼100 s of data acquisition, respectively, for shifts of the equilibrium coordinates for a significant 4 ∼3 × 10 samples of the photoionization of water. fraction of all the normal modes, which keeps nmax for Furthermore, these distance metrics are accompanied with most modes relatively low. Second, following sym- a success probability due to postselection of transmon heating metry considerations, there typically is structure in the events, which are 95% and 93% for the aforementioned Duschinsky rotation matrix which allows it to be approxi- simulations. The heating events are dominated by the coupler mated as block diagonal [34]. This is significant because transmon; the dynamics of a driven Josephson element for each block-diagonal subspace can be treated independently engineered bilinear operations presents a multidimensional as they do not couple to the remaining vibrational degrees optimization problem that seeks to maximize the desired of freedom of the molecule. This effectively reduces the interaction rates while minimizing induced decoherence and dimensionality of the problem from N to several subpro- blems with dimensionality of each block. In the limit where dissipation rates [31]P. Self-Kerr Hamiltonian terms of the ˆ † † no mode mixing occurs, i.e., U ¼ 1, the full multimode form H =ℏ ¼ − ∈ ðK =2Þcˆ cˆ cˆ cˆ , photon loss, Kerr i fA;Bg i i i i i Franck-Condon integrals can be easily and efficiently and imperfect state preparation account for errors that are computed via the product of all single-mode Franck- undetected by postselection. The first two effects are captured Condon integrals. through full time-domain master equation simulations (see Consequently, classical methods can implement conver- Supplemental Material [30]). The magnitudes of these two gence criteria that can provide a trade-off between accuracy errors depends on the molecular process; each corresponding and computational resources [35]. In cases where the Doktorov transformation will have different squeezing and Duschinsky rotation matrix cannot be well approximated rotation parameters thus leading to varying lengths of the as block diagonal, however, recursive methods inevitably pumped operations. Simulations of shorter length circuits need to explore the entire Hilbert space and therefore will therefore have lower error rates. Additionally, errors due become intractable for large system sizes. The advantage of to self-Kerr interactions of the cavities are larger for higher using a that samples from the full photon number states. distribution would then be to quickly identify the relevant FCFs with large weights, for which a classical computation VI. RESOURCE REQUIREMENTS can then be done for the specific FCFs of interest. Thus, the AND SCALABILITY utility of the quantum simulator for obtaining Franck- A. Classical methods Condon factors in large polyatomic molecules will depend on the configurational details for each electronic transition To properly contextualize the efficiency of the bosonic on a case-by-case basis. algorithm implemented in our work, it is useful to consider state-of-the-art computational methods for calculating B. Resource comparison: Bosonic versus Franck-Condon factors using classical computers. A perva- qubit processor sive andrepresentative approach is one that utilizes generating functions to compute a single Franck-Condon overlap inte- The central advantages of simulating the transformation of a bosonic Hamiltonian using a bosonic system lie in both gral hn1;n2; …;n jψ 0i in terms of other integrals with fewer N the native encoding and the efficient decomposition of occupations fhn1 − i1;n2 − i2; …;nN − iNjψ 0ig. This can be compactly represented through a recursion relation the Doktorov transformation into Gaussian operations. A recent proposal for obtaining Franck-Condon factors [32,33], which means that the total number of necessary 1 2 integrals to compute a Franck-Condon factor with a total on a conventionally envisioned spin- = quantum com- P puter [36] needs to map the problem onto qubits and a number of quanta M ¼ N n over N modes is [34] i¼1 i univeral gate set. This first requires encoding the Hilbert N N þ M − 1 space of size nmax onto nq ¼ Nlog2ðnmaxÞ qubits. The : ð12Þ M choice of nmax is dependent on the initial state as well as the magnitude of the displacement and squeezing; both

021060-7 CHRISTOPHER S. WANG et al. PHYS. REV. X 10, 021060 (2020) operations can produce states with large photon numbers. optimization for engineering the bilinear interactions [31] can Using quantum signal processing [37], the approximate reduce the error rate of the squeezing and beam splitter ˆ 10−3 number of gates ng then needed to implement UDok operations to , which increases the number of modes to using a universal qubit gate set to within an error ε is N ≈ 25. Beyond this, further reduction of the error rates or 2 2 3 1 ε implementing bosonic error correction protocols that preserve ng ¼ O½N nmax log ð = Þ. For our experiment with 2 16 bosonic statistics at the logical level [41] will be required for N ¼ modes, taking nmax ¼ and desiring an error −2 maintaining performance with increasing system size. ε ¼ 5 × 10 , this translates to nq ¼ 8 qubits and ng ¼ Oð103Þ gates. The coherence requirements for performing such a computation this way is thus relatively demanding VII. DISCUSSION AND OUTLOOK and exceeds the capabilities of current technologies, where The superconducting platform demonstrated here is we do not yet have fault tolerant quantum processors. By capable of successfully integrating all of the necessary comparison, our native bosonic simulator containing N components for performing a high-fidelity, scalable imple- modes simply requires 2N squeezing operations, N dis- mentation of a practical computational task of interest. placement operations, and a maximum of NðN − 1Þ=2 Looking ahead, there exist concrete steps toward scaling nearest-neighbor beam splitter operations [24,38]. This up, improving performance, and mitigating sources of 2 translates to a total of OðN Þ operations and a correspond- error. High-Q superconducting modules controlling up to ing circuit depth of OðNÞ when nonoverlapping beam three modes have been experimentally demonstrated [42], splitters are applied simultaneously. A possible advantage which would allow simulations to encompass nonlinear of the qubit-based algorithm, however, is the ability to triatomic molecules of Cs symmetry. In general, a molecule systematically incorporate anharmonicities in the PES, a composed of M atoms will have 3M − 6 vibrational task which still needs to be theoretically investigated for the degrees of freedom (3M − 5 for linear species), which sets bosonic implementation. the requirement for the number of modes needed in the simulator. The self-Kerr Hamiltonian terms of the cavity C. Scalability and error budget modes may be cancelled with extra off-resonant pump tones, or three-wave mixing methods that avoid self-Kerr As shown in Fig. 1, a linear array of bosonic memories nonlinearities altogether may be used [43]. with nearest-neighbor coupling is sufficient for scaling to To simulate a broader class of molecular systems, the larger system sizes. In our architecture of fixed frequency quantum simulator needs the capability of systematically cavity modes, the bilinear Gaussian operations are enacted incorporating inevitable deviations from harmonic behavior via robust frequency converting four-wave mixing proc- in the potential energy surfaces [19]. This is certainly an esses that obviate the need for any in situ frequency tuning. interesting and exciting direction to pursue in future As the hardware is scaled up, the fidelity of the experiments, especially given that classical computational individual operations will determine the fidelity of the methods have difficulty incorporating such anharmonic overall simulation. Though different photoelectron proc- effects even for relatively modest system sizes. Combining esses will have different errors, we can consider a sim- Josephson nonlinearities with external flux control, circuits plified model for quantifying performance with system with higher-order nonlinearities can be designed. These size. We associate a success probability for each operation nonlinear modes can then represent distinct vibrational fρ g where the index i encompasses state preparation (SP), i modes such that, when combined with bilinear [13] or displacements (D), squeezing (sq), beam splitters (BS), and higher-order interaction terms [44], a broader class of measurements (M). For simplicity, we assume a uniform anharmonic molecular Hamiltonians can be represented probability for each operation across all modes. We then accurately. More generally, for condensed matter many- specify a target success probability threshold ρ that th body systems, the tools utilized here enable the simulation reflects the accuracy of the full simulation. The number of tight-binding lattice Hamiltonians, and the inclusion of of modes that can be accurately simulated for a given ρ , th controllable self-Kerr interactions widens the scope to therefore, can be determined by extended Bose-Hubbard models [45,46]. The ability to use Josephson nonlinearities to manipulate microwave ρN ρ2NρNρNðN−1Þ=2ρN ρ SP sq D BS M > th: ð13Þ photons in these rich and diverse ways opens up promising avenues for the simulation of bosonic quantum systems in a Each of these probabilities can be taken to be the average superconducting platform. fidelity of each operation across a representative set of Doktorov transformations. Taking the expected bounds on ACKNOWLEDGMENTS the error rates of the operations due to photon loss in our experiment (Table I), while assuming a measurement error We thank Salvatore Elder, Hannah Lant, Zachary N. 10−2 ρ 0 5 ≈ 5 rate of and targeting th ¼ . ,wegetN .Modest Vealey, Lidor Foguel, Kenneth A. Jung, and Shruti Puri improvements in cavity lifetimes [39,40] and further circuit for helpful discussions. This research was supported by

021060-8 EFFICIENT MULTIPHOTON SAMPLING OF MOLECULAR … PHYS. REV. X 10, 021060 (2020) the U.S. Army Research Office (W911NF-18-1-0212, Q0 ¼ JQ00 þ K; ðA1Þ W911NF-16-1-0349). We gratefully acknowledge support of the U.S. National Science Foundation under Grants where Q0 and Q00 are mass-weighted normal coordinates of No. CHE-1900160 (V. S. B.), No. CHE-1464957 (P. H. V.), the pre- and post-transition molecular configurations, and No. DMR-1609326 (S. M. G.), the NSF Center for respectively. Because our simulation considers the trans- Ultracold Atoms (I. L. C.), and the Packard Foundation formation from a vibrational state in the pretransition (L. J.). Fabrication facilities use was supported by the Yale configuration to the post-transition configuration, we must Institute for Nanoscience and Quantum Engineering redefine the Duschinsky rotation matrices and associated (YINQE) and the Yale SEAS clean room. shift vectors accordingly: cos θ − sin θ APPENDIX A: OBTAINING U ¼ ¼ JT; ðA2Þ DOKTOROV PARAMETERS sin θ cos θ The Doktorov parameters originate from the physical d ¼ −JTK: ðA3Þ properties of a given molecule in the two electronic states of interest. Specifically, it is the structural information of the molecular configurations and the relationship between the 2. Conversion from molecular parameters two that fully parametrize the problem. to Doktorov parameters 1. Description of quantum-chemical analyses The Doktorov transformation as given in Eq. (2) of the main text is Theoretical predictions of optimized equilibrium ˆ † ˆ geometries (with imposed C2v symmetry constraints), Uˆ ¼ Dˆ ðαÞS ðζ0ÞRˆ ðUÞSðζÞ; ðA4Þ harmonic (normal-mode) vibrational displacements, and Dok Franck-Condon parameters (Duschinsky rotation matrices where for N ¼ 2 modes, the squeezing and displacement and associated shift vectors) exploited the commercial operations are defined as (G16 rev. A.03) version of the GAUSSIAN quantum-chemical suite (Table II) [47], with canonical Franck-Condon matrix ˆ † 0 ð†Þ ð0Þ ð†Þ ð0Þ Sð Þðζð ÞÞ¼Sˆ ðζ Þ ⊗ Sˆ ðζ Þ elements for specific vibronic bands being evaluated A 1 B 2 1 through use of the open-source ezSpectrum (version 3.0) ð0Þ 2 ð0Þ †2 ¼ exp ðζ cˆ − ζ cˆ Þ package [48]. All analyses relied on the CCSD(T) coupled- 2 1 A 1 A cluster paradigm, which includes single and double exci- 1 ⊗ ζð0Þ ˆ2 − ζð0Þ ˆ†2 tations along with noniterative correction for triples. exp 2 ð 2 cB 2 cB Þ ; ðA5Þ Dunning’s correlation-consistent basis sets [49–51] of triple-ζ quality augmented by supplementary diffuse func- ˆ ˆ ˆ D α D α1 ⊗ D α2 tions (aug-cc-pVTZ ≡ apVTZ) were deployed for all ð Þ¼ Að Þ Bð Þ α ˆ† − α ˆ targeted molecules except water, where a larger doubly ¼ expð 1cA 1cAÞ augmented, quadruple-ζ basis was employed (daug- ⊗ α ˆ† − α ˆ cc-pVQZ ≡ dapVQZ). expð 2cB 2cBÞ; ðA6Þ qffiffiffiffiffiffi The Duschinsky rotation matrices and associated shift 0 0 ζð Þ ν˜ð0Þ ν˜ð Þ vectors provided by the commercial package GAUSSIAN are where i ¼ lnð i Þ and i is the vibrational fre- defined via quency of mode i in the pretransition (post-transition)

TABLE II. Theoretically optimized molecular parameters. Vibrational frequencies for the symmetric-stretching and bending modes of ˜ −1 each molecule in pre- (ν˜) and post-transition (ν0) states are provided in wave numbers (cm ), which is related to angular frequency ω via ν˜ ¼ ω=2πc, where c is the speed of light. The rotation angle corresponding to the Duschinsky rotation matrix is defined in Eq. (A2). The shift vector K ¼ðk1;k2Þ is provided in mass weighted normal coordinates (where a0 is the Bohr radius and me is the mass) and reflects the relative displacement of equilibrium geometries between the two molecular configurations. pffiffiffiffiffiffi ν˜ −1 ν˜ −1 ν˜0 −1 ν˜0 −1 θ K Molecular photoelectron process stretch (cm ) bend (cm ) stretch (cm ) bend (cm ) (deg) ða0 meÞ ν h þ ˜ 2 − 3830.91 1649.27 2619.09 1602.85 −0.16598 (5.05, 49.47) H2O → H2O ðB B2Þþe − hν − 1031.10 582.58 1147.04 713.39 −0.0417 (27.36, 14.33) O3 → O3 þ e − hν − 1297.27 783.55 2633.34 796.94 2.401 46 (35.67, −38.01) NO2 → NO2 þ e ν h þ − 1136.38 506.27 1056.79 396.11 0.190 12 (−8.86, −58.34) SO2 → SO2 þ e

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pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 0 ffiffiffiffiffi 1 configuration and α ¼ ðω =2ℏÞd where fd g are the pν˜ 0 i i i i B 1 C vector elements of d in Eq. (A3). B C Ω B . . C The Duschinsky rotation matrix U generates the N-mode ¼ @ . A; ðA11Þ ffiffiffiffiffiffi rotation operator Rˆ ðUÞ. The multimode mixing elements p 0 ν˜N implemented in this experiment are two-mode beam splitters, necessitating a decomposition of U into near- 0 pffiffiffiffiffi 1 ˆ ν˜0 0 est-neighbor rotations, and thus R into nearest-neighbor B 1 C ˆ B C beam splitters. RðUÞ becomes a product of two mode beam Ω0 B . . C ¼ @ . A: ðA12Þ splitters parametrized by fθkg and fik;jkg, a sequence of pffiffiffiffiffiffi 0 ν˜0 angles andQ rotation axes derived from the decomposition N θ of U ¼ k Ri ;j ð kÞ. We can then write k k η Y The structure of L allows for a free scaling parameter † † which leaves L invarant; namely, Rˆ ðUÞ¼ exp½θ ðcˆ cˆ − cˆ cˆ Þ: ðA7Þ k ik jk ik jk k 0 Ω˜ ð Þ ¼ Ωð0Þ=η; ðA13Þ The decomposition of U is analogous to generalizing Euler Ω Ω0 Ω˜ Ω˜ 0 angles to SO(N); any rotation in RN can be written as a Lð ; Þ¼Lð ; Þ: ðA14Þ product of rotations in a plane R ðθ Þ, known as Givens ik;jk k Given that the squeezing operations of the Doktorov rotations. Following an algorithm similar to that in ζ Ω⃗ Refs. [24,38], but simplified to real orthogonal matrices, transformation take ¼ lnð Þ as inputs, an optimization produces a decomposition of U as a product of nearest- may be performed, as done in Ref. [18], that minimizes the neighbor rotation matrices. The Duschinsky matrix for total amount of squeezing while leaving the unitary N ¼ 2 is a single Givens rotation parametrized by an angle invariant. This is desirable as less squeezing corresponds θ which is enacted with one beam splitter: to shorter gate times in the simulation, which reduces the overall error rate. Table III lists the final set of dimension- Rˆ ( ˆ θ ) θ ˆ† ˆ − ˆ ˆ† less Doktorov parameters used in the experiment. Uð Þ ¼ exp½ ðcAcB cAcBÞ: ðA8Þ

APPENDIX B: THEORETICALLY PREDICTED 3. Optimization of squeezing parameters HAMILTONIAN TERMS The modification of the creation and annihilation oper- 1. Derivation of ancilla-mediated operations ators under the mode transformation is given in [16] In this Appendix, we derive the ancilla-mediated beam splitter and single-mode squeezing interactions as shown in 1 1 the main text as well as the associated ancilla-induced aˆ 0† ¼ ½L − ðLTÞ−1aˆ þ ½L þðLTÞ−1aˆ † þ α⃗; ðA9Þ 2 2 cavity frequency shifts. Derivations based on the perturba- tive four-wave frequency mixing enabled by a weak ancilla where nonlinearity have been presented previously in Ref. [13]. Here, we follow the formalism used in Ref. [31] and sketch L ¼ Ω0UΩ−1; ðA10Þ the general results without assuming weak ancilla non- linearity or weak pumps. We also give explicit expressions

TABLE III. Dimensionless Doktorov parameters. All values are truncated to the precision that the operations are able to be implemented experimentally. ν ν ν ν h − þ ˜ 2 − h − − h − h þ − H2O → e þ H2O ðB B2Þ O3 → O3 þ e NO2 → NO2 þ e SO2 → SO2 þ e

ζ1 0.262 0.104 0.035 0.242 ζ2 −0.160 −0.181 −0.217 −0.162 θ −0.166 −0.042 2.402 0.19 0 ζ1 0.072 0.157 0.389 0.206 0 ζ2 −0.174 −0.080 −0.208 −0.285 α1 −1.0162 −1.4278 0.0546 −0.1140 α2 −2.8977 −0.5311 −2.2207 1.7713 η 47.6381 28.9364 34.7639 26.4676

021060-10 EFFICIENT MULTIPHOTON SAMPLING OF MOLECULAR … PHYS. REV. X 10, 021060 (2020) for the strength of the engineered interactions in the case of transition frequencies. In particular, at zero pump ampli- weak pumps. tudes, the transmon’s transition frequency from the ground We start from the Hamiltonian of the two bare cavity to the first excited state decreases linearly with the cavity modes A and B coupled to the coupler transmon in photon number with a proportionality constant: module C: 2 δ χ δω − δω 2 gi i ∈ ˆ ℏ ω ˆ† ˆ ω ˆ† ˆ ˆ ˆ ˆ iC ¼ i;0 i;1 ¼ KC δ δ ;ifA; Bg; H= ¼ AcAcA þ BcBcB þ HC þ HI þ HpumpðtÞ: ðB1Þ i i þ KC ðB6Þ We emphasize that here the operators cˆA and cˆB are the annihilation operators for the bare cavity modes whereas 2 where δi ¼ ωi − ωC. Physically, the factor jgi=δij quan- in the main text the operators correspond to the dressed tifies the participation ratio of cavity A or B in the coupler cavity modes that are weakly hybridized with the ancilla transmon. In the experiment, this factor is 0.3% for cavity A transmons. ˆ and 0.2% for cavity B. HC is the Hamiltonian of the bare coupler transmon in Pumps on the coupler transmon can induce effective module C. After expanding the transmon potential energy inter- or intracavity interactions (single-mode squeezing or to fourth order in the phase across the Josephson junction beam splitter) denoted as Vˆ in Hˆ . For the case of the and neglecting counterrotating terms, we obtain [52] eff beam splitter interaction ðω2 − ω1 ¼ ωB − ωAÞ,wehave

† K †2 X Hˆ =ℏ ¼ ω tˆ tˆ − C tˆ tˆ2 ; ðB2Þ ˆ ℏ ˆ ℏ iφðmÞ ˆ ˆ† −iφðmÞ ˆ† ˆ C C C C 2 C C V= ¼ HBS= ¼ gBS;mðe cAcB þ e cAcBÞ m ˆð†Þ ⊗ Ψ Ψ where tC again is the bare annihilation (creation) operator j mih mj: ðB7Þ for the coupler transmon with frequency ωC and anharmo- ω ω 2ω nicity KC. For the case of single-mode squeezing ( 1 þ 2 ¼ A or ˆ 2ω ), we have HI is the interaction energy between the coupler trans- B mon and the two cavity modes. Neglecting counterrotating X iϕðmÞ 2 −iϕðmÞ †2 ˆ ℏ ˆ ℏ sq;i ˆ sq;i ˆ terms, it may be written as V= ¼ Hsq;i= ¼ gsq;i;mðe ci þ e ci Þ m † Hˆ =ℏ ¼ðg cˆ þ g cˆ Þtˆ þ H:c: ðB3Þ I A A B B C ⊗ jΨmihΨmj;i∈ fA; Bg: ðB8Þ ˆ HpumpðtÞ represents two pumps on the coupler transmon: Similar to the transmon-induced frequency shifts on the † cavities, here both the strength and phase of the transmon- ˆ ℏ Ω −iω1t Ω −iω2t ˆ HpumpðtÞ= ¼ð 1e þ 2e ÞtC þ H:c: ðB4Þ mediated interactions depend on the state of the transmon. Of primary interest to us is the strengths of the transmon- Of primary interest to us is the dispersive regime where mediated interactions when the transmon is in the Floquet the cavity-transmon coupling strengths are much smaller state jΨ0i. For weak drives, these strengths are than their detuning: jg j ≪ jω − ω j [53]. In this A;B A;B C regime, we can treat the cavity-transmon interaction as a Ω Ω δ δ ≈ 2 gA gB 1 2 A þ 2 perturbation (while treating the remaining parts of the gBS;0 KC δA δB δ1 δ2 δA þ δ2 þ KC Hamiltonian exactly), and to second order in the interaction s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi strength, we obtain an effective Hamiltonian after perform- ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi δ δ pχ χ ð A þ KCÞð B þ KCÞ ing a unitary on Hˆ ¼ AC BC δAδB X † † Ω Ω δ δ ˆ ℏ δω ˆ ˆ δω ˆ ˆ ⊗ Ψ Ψ ˆ 1 2 A þ 2 Heff= ¼ ð A;mcAcA þ B;mcBcBÞ j mih mjþV; × ; ðB9Þ m δ1 δ2 δA þ δ2 þ KC ðB5Þ 2 Ω Ω δ ≈ 2 gi 1 2 i gsq;i;0 KC Ψ δ δ1 δ2 2δ þ K where j mi is the mth Floquet state that quasiadiabatically i i C connects to the mth Fock state jmi of the bare transmon Ω Ω δ χ 1 2 i þ KC ∈ as the pumps are ramped up or down [31]. At zero pump ¼ iC ;ifA; Bg; ðB10Þ δ1 δ2 2δi þ KC Ψ ˆ amplitudes, j mi¼jmi. The first term in Heff thus represents transmon-induced cavity frequency shifts where δ1;2 ¼ ω1;2 − ωC. In the case where the transmon δω δω Ψ A;m and B;m when the transmon is in j mi. anharmonicity KC is small compared to the detunings δω δω The difference between A;m or B;m with different jδ1;2;A;Bj, the expressions above reduce to those obtained m leads to cavity-photon-number-dependent transmon based on perturbative multiwave frequency mixing [13].

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Note that the interaction strengths presented in the main where fðxÞ¼ð18x3 þ 30x2 þ 22x þ 6Þ=½4ðx þ 1Þð4x2þ 8 3 χ0 text and the rest of the appendixes refer to the values of x þ Þ. We note that there is also a contribution to iC gBS;0 and gsq;i;0. from a term in the sixth-order expansion of the transmon For weak drives, Eqs. (B9) and (B10) show that the cosine potential, but for ωC ≫ jδij this correction is strengths of the engineered beam splitter and single-mode negligible. squeezing increase linearly with both drive amplitudes Here we have only considered the cavity Kerr induced by Ω1;2. For strong drives, this dependence becomes nonlinear the coupler transmon. In general, the transmon ancillas in in Ω1;2 and can be accurately captured using Floquet theory modules A and B also induce Kerr in their respective for the driven transmon [31]. We have verified that the cavities. The total Kerr of each cavity will then be the sum experimentally measured beam splitter and single-mode of all contributions. squeezing rates match the expressions (B9) and (B10) for In the presence of pumps on the coupler transmon, the weak drives and the full Floquet analysis at strong drives. cavity Kerr can be strongly modified due to a relatively strong hybridization between cavity photons and excita- tions of the coupler transmon. To illustrate this effect, 2. Transmon-induced cavity Kerr we consider as an example the pumps used in generating Another important effect and a source of infidelity is the the beam splitter interaction between the two cavities. For cavity nonlinearity induced by the transmons. To fourth the choice of pumps used in the experiment, the sum of the order in the cavity-transmon coupling, this nonlinearity is a frequency of cavity A and the higher-frequency pump is Kerr nonlinearity and has the following form: close to the frequency of transition from transmon ground to the second excited state: ωA þ ω2 ≈ ω02. As a result, the X K †2 K †2 cavity photons become relatively strongly hybridized with Hˆ ℏ − A;m ˆ ˆ2 − B;m ˆ ˆ2 Kerr= ¼ 2 cA cA 2 cB cB the second excited state of the transmon, thus modifying m their nonlinearity. Using a sixth-order perturbation theory † † (fourth order in g and second order in Ω2), we find that the − K cˆ cˆ cˆ cˆ ⊗ jΨ ihΨ j; ðB11Þ A AB;m A A B B m m modification to the cavity Kerr is

Ω 2 χ2 2δ δ where KA;m and KB;m are the self-Kerr of cavities A and B δ ≈ 2 2 AC ð 2 þ KCÞ 2 KA;0 KC 2 ; ðB15Þ and KAB;m is the cross-Kerr between cavities A and B when δ2 Δ ðδ2 þ KCÞðδ2 − KCÞ the transmon is in the state jΨmi. First, we consider the case in the absence of pumps. Of where Δ ¼ ωA þ ω2 − ω02, and ω02 is the Stark shifted interest to us is the cavity Kerr when the transmon is in the transmon transition frequency from the ground to the ground state j0i: second excited state. The above expression, which applies for small jΔj, qualitatively captures the observed enhanced 4 δ χ2 δ 2 self-Kerr of cavity A in the experiment during the beam 2 gi i iC ð i þ KCÞ Ki;0 ¼ KC ¼ ; splitter operation. Comparing this expression with that of δi 2δi þ KC 2KC δið2δi þ KCÞ the bare cavity Kerr KA;0 without pumps, we see that δKA;0 i ∈ fA; Bg; ðB12Þ becomes comparable to KA;0 when KCjΩ2=δ2j ∼ jΔj. We note that such dependence of the cavity Kerr on the 2 δ δ drive parameters also potentially provides a knob to control 2 gA gB KCð A þ BÞ KAB;0 ¼ the cavity Kerr for the purpose of simulating nonlinear δ δ δ þ δ þ K A B A B C bosonic modes. χ χ ðδ þ K Þðδ þ K Þ δ þ δ þ B ¼ AC BC A C B C A : 2K δ δ δ þ δ þ K C A B A B C APPENDIX C: SYSTEM CHARACTERIZATION ðB13Þ 1. Calibration of Gaussian operations In the dispersive regime, the transition frequency ωl of Also of interest to us is the difference between Ki;0 and ti ancilla tˆ depends on the photon number l in the respective Ki;1. This difference leads to a nonlinear dependence of the i transmon transition frequency on the cavity photon number. cavity: This difference is usually denoted as χ0 ωl ¼ ω0 − lχ þðl2 − lÞ i ; ðC1Þ 2 ti ti i 2 K 0 − K 1 χ χ0 i; i; iC δ iC ¼ 2 ¼ δ fð 1=KCÞ; i ω0 where ti is the ancilla frequency when there are no photons i ∈ fA; Bg; ðB14Þ χ χ0 in its respective cavity and i and i are the dispersive shifts

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(a) (b) (c) Single photon probability Photon number probability Photon number probability

Calibrated Squeezing duration (μs ) Beam splitter duration (μs)

FIG. 4. Calibrations for Gaussian operations. (a) Displacement calibrations for cavities A (top) and B (bottom) starting in vacuum. Here, the amplitude of a resonant pulse is varied. (b) Squeezing calibration for cavities A (top) and B (bottom) starting in vacuum. Here, the length of two squeezing pump tones is varied. Legends indicate population in photon number n. (c) Beam splitter calibration for beginning with a single photon in cavities A (top, jψ 0i¼j1; 0i) and B (bottom, jψ 0i¼j0; 1i). The length of two beam splitter pump tones is varied, and the probability that the photon remains in the cavity that it started in is plotted over time. originating from fourth- and sixth-order Hamiltonian terms, 2. Measurement of system parameters respectively, as introduced in the main text. Using this, the Static and pump-induced self-Kerr Hamiltonian terms, photon number population of each cavity can be extracted †2 2 †2 2 −ðK =2Þcˆ cˆ and −ðK =2Þcˆ cˆ , are estimated using the via π pulses selective on each photon number after applying A A A B B B protocol detailed in Ref. [54] (Fig. 5). For the pump- various strengths of each operation (Fig. 4). These popu- induced cases, one of the two pumps is detuned by δ ¼ 20 lations are then fit to the corresponding expected models, and 50 kHz for squeezing and beam splitter operations, including an overall offset and scaling factor to take into respectively, to make the engineered interaction off reso- account errors due to ancilla relaxation and readout imper- nant. We assume that the induced self-Kerr is not a strong fections (Table IV). For the beam splitter, we assume an function of this detuning. effective detuning between cavities A and B in a frame where δ 0 Static and pump-induced cavity decay rates are mea- BS ¼ if the beam splitter resonance condition is satisfied. sured via T1 experiments (Fig. 6). A single photon is Transmon heating leads to fluctuations in δ , which BS prepared in each cavity, followed by either a delay or an off- dephases the beam splitter operation with a dephasing rate: R resonant pumped operation (with the same detunings as κBS ∞ δ t δ 0 dt. Thus, the oscillating popula- ph ¼ 0 h BSð Þ BSð Þi above). Again, we assume that the pump-induced decay tions of a single photon in each cavity P10=01 is given to rates are not a strong function of the pump detuning. The κ κBS leading order in A;B=gBS and ph =gBS by the expression in ancillas are then flipped via selective π rotations condi- κ¯ κBS κBS 2 κBS 1 Table IV, where ¼ð A þ B Þ= and A;B are the effective tioned on n ¼ photon. In both cases, the data are linewidths of cavities A and B during the beam splitter postselected on the ancilla being in the ground state before operation. the selective π rotation. We attribute the higher decay rate to

TABLE IV. Calibrated rates of Gaussian operations. The amplitude of a displacement operation of fixed length τ ¼ 72 ns is calibrated for generating a with α ¼ 1 [Fig. 4(a)]. The rates for the squeezing and beam splitter operations are extracted from the fits [Figs. 4(b) and 4(c)].

Operation Model Cavity Calibrated rate 2l 2 τ 72 Displacement PðlÞ¼ðjαj =l!Þ expð−jαj Þ A α¼1 ¼ ns τ 72 B α¼1 ¼ ns 2l 2 2l ≈ 60 Squeezing Pð2lÞ¼½ð2lÞ!=2 ðl!Þ ½tanh ð2gsqtÞ= coshð2gsqtÞ Agsq kHz Bgsq ≈ 60 kHz BS 1 −κ¯ − −κ t−t0 =2 Beam splitter ðt t0Þ 1 ph ð Þ 2 − A P10=01 ¼ 2 e × f þ e cos½ gBSðt t0Þg g ≈ 2π × 44 kHz B BS

021060-13 CHRISTOPHER S. WANG et al. PHYS. REV. X 10, 021060 (2020)

(a) (b) (c) Frequency (kHz)

Average photon number

FIG. 5. Estimation of intrinsic and pump-induced self-Kerr. In all plots, the y coordinate corresponds to the effective frequency of a coherent state with average photon number n¯ on the x coordinate. The slope determines the self-Kerr, and the offsets reflect pump- induced Stark shifts. Experiments for estimating the self-Kerr in the (a) absence of pumps, (b) presence of off-resonant squeezing pumps, and (c) presence of off-resonant beam splitter pumps for cavity A (top panels) and B (bottom panels).

the hybridization of the cavities with the shorter-lived coupler transmon. Measured cavity Kerr and T1 values are given in Table V.

APPENDIX D: CIRCUIT IMPLEMENTATION The full implemented in our experiment is shown in Fig. 7. State preparation in our experiment probability

Single photon (Fig. 8) consists of first performing measurement-based feedback cooling of all modes to their ground state (this protocol is described in full detail in the Supplemental Material of Ref. [10]). For preparing Fock states, optimal control pulses are then played that perform the following Delay (μs) state transfers:

FIG. 6. Measurement of intrinsic and pump-induced decay j0i ⊗ jgi → jni ⊗ jgi ; rates. Cavity T1 experiments with either a varying delay (red) or A tA A tA the application of an off-resonant squeezing operation (purple) 0 ⊗ → ⊗ j iB jgit jmiB jgit : ðD1Þ during the delay for cavities A (top) and B (bottom). B B

These state transfers, however, suffer a finite error prob- ability on the order of a few percent due to decoherence TABLE V. Estimated cavity Kerr and T1 values. The beam during the operation. This error is suppressed by perform- splitter decay rates are extracted from the fit performed in the ing a series of QND measurements of each cavity photon calibration of the operation in Fig. 4(c) assuming that number and postselecting on outcomes that verify that the κBS κBS κ¯. A ¼ B ¼ correct state was prepared. This is done via k selective π

Cavity Operation K=2π (kHz) T1 (μs) rotations on the ancilla transmons conditioned on the desired photon numbers in the cavities, followed by A Native 1.8 280 transmon measurements—even if the desired state is joint Squeezing 2 200 Beam splitter 30 170 vacuum. The final data are postselected on the ancilla measurement outcomes being ð“e; ”“g”Þ⊗k=2 for both B Native 3.2 320 modules, where k is chosen to be even. In our experiment, Squeezing 1.9 280 we choose k ¼ 2 for the single-bit extraction measurement Beam splitter 5 170 scheme and k ¼ 6 for the sampling measurement scheme.

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(a) (b) Ancilla A Cavity A Cavity A

Coupler Coupler

Cavity B Verification

Initialization Cavity B Measurement Ancilla B

(c) (d) i = 0,1,2,3 Ancilla A Z Ancilla A Z

Cavity A Cavity A n' bi m' b Cavity B Cavity B i

Ancilla B Z Ancilla B Z

FIG. 7. Circuit implementation of the Franck-Condon simulation. (a) Overview of the quantum simulation algorithm, consisting of state preparation, unitary Doktorov transformation, and measurement. A set of verification measurements is performed after the unitary Doktorov transformation for the purpose of postselecting the final data on measuring the transmons in their ground state. (b) The two- mode circuit decomposition of the Doktorov transformation used in this experiment. The nonlinearity of the coupler transmon is primarily utilized to perform all three pumped operations, though in principle that of the ancilla transmons could have been used as well. 0 0 (c) Single-bit extraction. Selective π pulses ðRπÞ flip each ancilla transmon conditioned on having n and m photons in cavities A and B, respectively, for a given run of the experiment. The ancillas are then simultaneously read out using standard dispersive techniques. 0 0 Subsequent runs of the experiment thus need to scan n and m over the photon number range of interest up to the desired nmax. (d) Sampling. Optimal control pulses are designed to excite each ancilla transmon from jgi to jei conditioned on the value of the binary bits bi of each cavity state, followed by dispersive readouts. Here, we measure the first 4 bits on a given run of the experiment, thus resolving the first 16 Fock states for each cavity. Real-time feed-forward control is used to dynamically reset the state of the ancilla in between bit measurements to minimize errors due to ancilla relaxation.

k APPENDIX E: CORRECTING SYSTEMATIC Ancilla A Z ERRORS DUE TO TRANSMON DECOHERENCE DURING SINGLE-BIT EXTRACTION Cavity A prepare n Errors due to ancilla decoherence during the single-bit extraction measurement scheme may be systematically m Cavity B calibrated out. Specifically, decay and heating events during selective π rotations and readout errors result in a systematic bias in the final estimate of the photon number Ancilla B Z

prepare population. For the case of a single ancilla qubit coupled to a single cavity, these effects result in a reduction of contrast for a Rabi experiment when both the ancilla and the cavity FIG. 8. Circuit implementation of heralded state preparation. are prepared in their ground state (Fig. 9). Measurement-based feedback cooling techniques prepare the full 0 When using this pulse to infer cavity photon number system in its ground state (i.e., both cavities in j i and all populations, we assume that there is no photon number transmons in jgi). Optimal control pulses of the form listed in Table I of the main text are played simultaneously on each dependence to either the Rabi or decoherence rates of the module to prepare a desired photon number state, followed by a ancilla. Under this model, we can relate the measured set of k check measurements. probabilities Q⃗to the true probabilities P⃗via

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