Quantum-enhanced bosonic learning machine

Machine learning techniques excel at finding patterns in data that are usually encoded as high dimensional feature vectors. As the size of these vectors grows, processing them classically requires enormous computational resources Lloyd et al. (2013). Recently proposed quantum algorithms show evidences of computational speedups, that comes from the exploitation of large Hilbert space of quantum systems Lloyd et al. (2013, 2014, 2016); Rebentrost et al. (2014). Demonstrations of quantum machine learning algorithms implemented on discrete variables platforms have shown tremendous potential Cai et al. (2015); Trávníček et al. (2019); Havlíček et al. (2019); Johri et al. (2020); Bartkiewicz et al. (2020); Xin et al. (2021). However, the number of gates and qubits must be increased for solving practical problems on near-term quantum processors.

An alternative approach is to utilize the infinite-dimensional Hilbert space of bosonic systems as the feature space for quantum machine learning Lau et al. (2017); Schuld and Killoran (2019). One advantage of the bosonic architecture is that a single operation, such as squeezing or displacement, is formally equivalent to processing infinitely-dimensional feature vectors Schuld and Killoran (2019). Furthermore, it allows for efficient use of physical resources. For discrete-variable schemes, approximately $\log(n)$ physical qubits are required to encode an $n$-bit data string, whereas in the alternate approach, one bosonic mode is sufficient. As an example, in the ideal case the motion of a single trapped ion suffices to encode 3 infinitely dimensional feature vectors, whereas the internal state of an ion typically only encodes a single qubit Ortiz-Gutiérrez et al. (2017).

In this letter, we report a bosonic learning machine that consists of two building blocks for encoding and processing either quantum or classical data (see Fig. 1d). First, a universal embedding circuit implements a feature map $\phi$ of classical data $\boldsymbol{x}=(x_{1},x_{2},...,x_{n})$ to a quantum state $|\phi(\boldsymbol{x})\rangle$. The map chosen in this work is amplitude encoding $|\phi_{\textrm{amp}}(\boldsymbol{x})\rangle=1/||\boldsymbol{x}||_{2}\sum\limits_{j=1}^{n}x_{j}|j-1\rangle$, where $\{|j\rangle\}$ are the Fock states, $n$ denotes the dimension of the data, and $||\boldsymbol{x}||_{2}=\sqrt{\sum\limits_{j=1}^{n}x^{2}_{j}}$ is the Euclidean norm. The universal embedding circuit $\phi$ can be associated with a state preparation gate sequence $U_{\phi}(\boldsymbol{x})$, that prepares an arbitrary state $|\phi(\boldsymbol{x})\rangle$ satisfying $U_{\phi}(\boldsymbol{x})|0\rangle=|\phi(\boldsymbol{x})\rangle$ Schuld and Killoran (2019); Lloyd et al. (2020).

Machine learning algorithms frequently rely on distance measures that can quantify the similarity between feature vectors Hastie et al. (2009). Here, we employ the Hilbert-Schmidt distance defined as $D_{\textrm{HS}}(|\phi(\boldsymbol{x})\rangle,|\phi(\boldsymbol{x^{\prime}})\rangle)=\sqrt{2-2|\langle\phi(\boldsymbol{x})|\phi(\boldsymbol{x^{\prime}})\rangle|^{2}}$, where $|\langle\phi(\boldsymbol{x})|\phi(\boldsymbol{x^{\prime}})\rangle|^{2}$ is the overlap between two quantum states $|\phi(\boldsymbol{x})\rangle$ and $|\phi(\boldsymbol{x^{\prime}})\rangle$ Trávníček et al. (2019). The overlap measurement, reported throughout this work, can be efficiently realized with a constant-depth circuit implemented by a SWAP test Filip (2002); Gao et al. (2018); Gan et al. (2020); Nguyen et al. (2021).

In this work, the bosonic learning machine comprises two trapped ¹⁷¹Yb⁺ ions in a linear radio-frequency Paul trap, employing three motional modes denoted $A$, $B$, and $C$. We start the experiment by preparing all the modes in the ground state of motion. Two of the modes, $B$ and $C$, are used for storing information while $A$ acts as an auxiliary mode required for implementing the SWAP test. The hyperfine states $|g\rangle,|e\rangle$ of one ion mediate interaction between the motional modes, while the other ion is prepared in the ${}^{2}F_{7/2}$ metastable state and does not interact with optical fields during the experiment (See Fig.1). The Jaynes-Cumming interaction between the internal state and the motional modes enables deterministic preparation of an arbitrary state with dimension $n$ in $n-1$ steps, using methods presented in Law and Eberly (1996); Ben-Kish et al. (2003). Alternatively, we use displacement and squeezing operations to prepare coherent and squeezed vacuum states respectively. The SWAP test relies on controlled beam splitter gates of the form $U_{\textrm{CBS}}(\theta,\psi)=\exp[-(i/\hbar)\int_{0}^{T}\sigma_{x}H_{\textrm{BS}}(\psi,t)dt)]$ that enacts the Hamiltonian $H_{\textrm{BS}}(\psi)/\hbar=\frac{g(t)}{2}(a^{\dagger}be^{i\psi}+ab^{\dagger}e^{-i\psi})$, depending on the internal state of the ions in the $\sigma_{x}$ basis Filip (2002); Gao et al. (2018); Nguyen et al. (2021). Here $a(a^{\dagger})$ and $b(b^{\dagger})$ are the annihilation (creation) operator of the motional modes, $g(t)$ is the coupling strength, $\psi$ an experimentally tunable phase, and $\theta=\int_{0}^{T}g(t)dt$ the effective mixing angle for a gate duration $T$ (Methods).

Given two quantum states $|\phi(\boldsymbol{x})\rangle_{B}$ and $|\phi(\boldsymbol{x^{\prime}})\rangle_{C}$ prepared in mode $B$ and $C$ respectively, the SWAP test can be implemented with controlled beam splitter gates $U_{\textrm{CBS}}(\pi,0)$ acting between modes $A$ and $B$, and $U_{\textrm{CBS}}(\pi/2,\pi)$ between modes $A$ and $C$, followed by a projection measurement of the internal state. The overlap between two states is proportional to the probability $P_{g}$ to detect the internal state in $|g\rangle$: $|_{B}\langle\phi(\boldsymbol{x})|\phi(\boldsymbol{x^{\prime}})\rangle_{C}|^{2}=|1-2P_{g}|$ (Fig.1d). The SWAP test duration is independent from the dimension of the data and is significantly shorter than the coherence times of the motional modes Nguyen et al. (2021).

To highlight applications of our bosonic learning machine, we demonstrate the unsupervised $K$-means clustering and supervised $k$-NN algorithms using quantum states as the input data. We note that the input data may also be classical as it can be mapped to quantum states by using the universal embedding circuit. We begin with the description of implementing the unsupervised $K$-means clustering algorithm. Consider two parties Alice and Bob. Alice prepares and sends Bob identical copies of $M$ quantum states. The quantum states can be expressed in the Fock basis $|d_{m}\rangle=\sum\limits_{j=0}^{n-1}x_{m,j+1}|j\rangle$, where $m$ runs from $1$ to $M$. In this scenario, Bob is assumed to have no a priori information about the states that Alice prepares. Without resorting to a full tomography of the input states, Bob’s task is to sort the quantum states into $K$ clusters based on similarities among the states.

The algorithm begins with Bob making a guess of the number of clusters $K$, and associates each cluster to a centroid state that are randomly initialized. The states are assigned to the cluster that has the nearest centroid. In step $l$ we implement a map $|d_{m}\rangle\rightarrow r^{m}_{l}=\operatorname*{argmin}_{k}\{D(|d_{m}\rangle,|c_{lk}\rangle)\}$, where $|c_{lk}\rangle$ denotes the centroids corresponding to the $k-$th cluster. Our quantum advantage comes from the reduction of complexity involved in estimating the distance of the two states $|d_{m}\rangle$ and $|c_{lk}\rangle$. This estimation takes approximately $n$ steps to perform classically, in comparison to the constant-time operation in our experiment. The centroids are subsequently updated by Alice to be the mean of the states contained in the corresponding cluster. The assignment and update are repeated until the clusters cease to change.

Our data is a set of 15 quantum states and consists of three clusters: the set $\mathcal{H}_{\textrm{sqz}}$ of squeezed vacuum states $|re^{i\pi}\rangle$, $\mathcal{H}_{\textrm{Fock}}$ superposition of Fock states $\cos(\varphi/2)|0\rangle+\sin(\varphi/2)|1\rangle$, and $\mathcal{H}_{\textrm{coh}}$ coherent states $|\alpha\rangle$. We choose the parameters as $r\in\{1.5\,,1.1\,,1.2\,,1.3\,,1.4\}$, $\varphi/\pi\in\{0.37\,,0.45\,,0.47\,,0.72\,,0.55\}$, and $\alpha\in\{1.5\,,1.3\,,1.0\,,1.2\,,1.1\}$. Our simulations verify that $K$-means algorithm can partition this data set correctly on a classical computer. The data states are prepared in the mode $B$. For the first step, we choose the centroid states in mode C to be $|c_{01}\rangle=|0\rangle$, $|c_{02}\rangle=|1\rangle$, and $|c_{03}\rangle=|2\rangle$. After each step the centroids, truncated to dimension 5, are updated and approximately synthesized using the embedding circuit. We characterize the state generation through measurements of Rabi oscillations on the $|g\rangle|n+1\rangle$ to $|e\rangle|n\rangle$ red-sideband transition (see Methods).

Figure 2a shows the overlap measurements between the data states and the centroids. We used on average 700 experimental runs for each measurement. With this particular choice of the initial centroids, the algorithm attains a stationary assignment after four steps. The final classification result is given in Fig. 2b, where the scatter plot of the amplitudes $x_{j}$ of the data states are shown for $j$ up to 5. The algorithm associates cluster 1 to $H_{\textrm{sqz}}$, cluster 2 to $H_{\textrm{Fock}}$, and cluster 3 to $H_{\textrm{coh}}$. The SWAP test measurements, albeit imperfect, produce classification results that agrees perfectly with the prior knowledge about the states sent to Bob. Stationary assignment of the centroids is evident from the step-by-step evolution shown in the inset of Fig. 2b. The assignment results after each step can be found in Extended Data.

We next demonstrate the $k$-NN algorithm to classify unknown trial states, by using the three labelled clusters from the $K$-means results as training states. The traditional approach to solving the problem of classifying unknown states is to perform a complete tomography, which requires a number of measurements that grows exponentially with the state dimension $n$ Haah et al. (2017). However, if complete information regarding the state is unnecessary, $k$-NN algorithm could offer an efficient alternative, especially when $n$ is large. With a constant depth SWAP test, computational complexity of the $k$-NN becomes independent of $n$.

The $k$-NN algorithm requires estimation of the overlap between each trial state and all training states. We identify the $k$ states having highest overlap with the trial state. The trial state is then assigned to the cluster that labels the majority of $k$ states. Figure 3 shows the experimental results of $k$-NN classification with $k=7$ for a variety of trial states. The first three trial states utilize the same operation that prepares the three clusters, respectively, and are observed to be correctly classified. We further carry out the classification of trial states that are prepared with other operations (See Methods). Such examples are the two states $D(\alpha=1)(|0\rangle-|1\rangle)$ and $\sqrt{0.3}S(r=0.8e^{i\pi})|0\rangle+\sqrt{0.2}(|0\rangle+|1\rangle)+\sqrt{0.5}D(\alpha=1.7)|0\rangle$ (up to a normalization factor), that are both classified to cluster 2. These results indicate that the outcome of the algorithm is determined by overlap of the quantum states, which can be visualized by the similarity of their Wigner functions. This can be interpreted as a form of Wigner function pattern recognition, much like applying $k$-NN to handwriting recognition in classical ML Hastie et al. (2009).

In conclusion, we realize a bosonic learning machine using motional modes in a system of trapped ions. The core subroutine of the machine is the efficient SWAP test that estimates the overlap of two motional modes with a constant gate duration, a feature that is preferred in solving problems with large dimension such as quantum support vector machines. We expect that the experimentally accessible dimensions can be further increased by improving various technical aspects of the setup, such as the motional coherence time and coupling strength between modes. The same bosonic learning machine can also be used to implement several quantum kernels proposed in Schuld and Killoran (2019) and on different experimental platforms such as superconducting microwave cavities Hofheinz et al. (2009); Gao et al. (2018). These results provide a key primitive for developing advanced quantum-enhanced machine learning algorithms in infinite-dimensional feature spaces.

Author contributions

C.H.N. and D.M. conceived the experiment. C.H.N. K.W.T, G.M. and H.C.J.G. built and maintained the experimental apparatus. C.H.N. and K.W.T performed the experiments and analyzed the experimental data with assistance from G.M. and H.C.J.G. All authors contributed to the manuscript preparation.

Data availability

The data that support the findings of this study are available from the corresponding authors on reasonable request.

Author Information

The authors declare no competing financial interests. Correspondence and requests for materials should be addressed to C.H.N. ([email protected]) or D.M. ([email protected]).

Methods

Experimental setup. We trap two ¹⁷¹Yb⁺ ions in a four-rod linear radio-frequency Paul trap with single-ion secular trap frequencies $(\omega_{x},\omega_{y},\omega_{z})=2\pi\times(1.274,0.945,0.519)$ MHz. The normal motional modes $A$, $B$, and $C$ are the out-of-phase motion in the $y$ direction, out-of-phase motion and in-phase in the $x$ direction with frequencies $(\omega_{A},\omega_{B},\omega_{C})=2\pi\times(0.782,1.159,1.274)$ MHz, respectively. The transverse trapping frequencies are actively stabilized and drift less than 100 Hz/hour.

One of the ions, pumped to the metastable state ${}^{2}F_{7/2}$ by driving the ${}^{2}D_{3/2}\rightarrow$ $3[1/2]^{\circ}_{3/2}$ transition at $398.98$ nm, does not interact with optical fields and is referred to as the “dark” ion. At the beginning of all experiments, the other ion is Doppler cooled by a 369.53  nm laser red-detuned from the $|^{2}S_{1/2},F=1\rangle$ to $|^{2}P_{1/2},F=0\rangle$ transition, and sympathetically cools the “dark” ion. The two hyperfine states used in the experiment are $|g\rangle=|^{2}S_{1/2},F=0,m_{F}=0\rangle$ and $|e\rangle=|^{2}S_{1/2},F=1,m_{F}=0\rangle$, which are separated by $\omega_{0}/2\pi=12.643$ GHz. The ion is initialized via optical pumping in the state $|g\rangle$. Resonance fluorescence state detection technique is used to detect the ion’s internal state Olmschenk et al. (2007).

A frequency-doubled, mode-locked Ti:Sapphire laser with a central wavelength of 372.85 nm is used to drive the stimulated Raman transitions. The mode-locked laser has a pulse duration of 3 ps and a repetition rate of 76.2 MHz. Two beams, R1 and R2 (Fig. 1a,b) with an average power of 60mW each, are sent through acoustic-optical modulators (AOMs) and focused to a beam waist of $\sim 15\,\mu$m at the ions. The beams R1 and R2 form 45^∘ and 135^∘ angles with respect to the $z$-axis (Fig. 1a). The detuning of the Raman beams are controlled by the AOMs.

We begin all experiments with all the radial motional modes prepared in the ground state (residual $\bar{n}\leq 0.05$) using 10 ms Sisyphus cooling followed by 250 cycles of Raman sideband cooling.

Controlled beam splitter gates. We implement controlled beam splitter gates $U_{\textrm{CBS}}$ between modes $A$ and $B$ by applying a bichromatic Raman beatnote at $\omega_{0}\pm\omega_{AB}$, where $\omega_{AB}=|\omega_{A}-\omega_{B}|$. The resulting Hamiltonian is

where $\psi=\frac{\psi_{b}-\psi_{r}}{2}$, and $\psi_{r}$ and $\psi_{b}$ are the phases of the bichromatic fields. We let $\sigma_{x}=(e^{-i\psi_{S}}\sigma_{+}+e^{i\psi_{S}}\sigma_{-})$, where $\psi_{\textrm{S}}=-\frac{\psi_{r}+\psi_{b}}{2}$, and $\sigma_{+}=|e\rangle\langle g|$, $\sigma_{-}=|g\rangle\langle e|$. $g(t)$ is the coupling strength.

The controlled beam splitter gate corresponds to the unitary evolution $U_{\textrm{CBS}}(\theta,\psi)=\exp[-(i/\hbar)\int_{0}^{T}\sigma_{x}H_{\textrm{BS}}(\psi,t)dt)]$, where $\theta=\int_{0}^{T}g(t)dt$. In the experiment, we obtain a coupling strength $g/2\pi\sim 680$ Hz. This corresponds to implementing a beam splitter operation $U_{\textrm{CBS}}(\theta=\pi/2)$ in $T\sim 365$ $\mu$s, much shorter than the motional phase coherence time for the superposition of $|0\rangle+|1\rangle$ which is approximately $10$ ms.

Preparation and characterization of quantum states. The universal embedding circuit relies on a series $n-1$ pairs of carrier and red-sideband:

where $t_{l}$ ($\tau_{l}$) and $\upsilon_{l}$ ($\Upsilon_{l}$) are the duration and phase for the $l$-th carrier (red-sideband) pulse that are determined by the Eberly-Law algorithm Law and Eberly (1996). The additional phases $\upsilon^{{}^{\prime}}_{l}$ compensate for the rotation of the internal states due to the AC Stark shifts of the red-sideband pulses (See Supplementary). The embedding circuit is used to prepare the centroid states of $K$-means, the cluster state $\mathcal{H}_{\textrm{Fock}}$ of $K$-means experiment, and the trial state (up to a normalization factor) $\sqrt{0.3}S(r=0.8e^{i\pi})|0\rangle+\sqrt{0.2}(|0\rangle+|1\rangle)+\sqrt{0.5}D(\alpha=1.7)|0\rangle$ of $k$-NN experiment. We use spin-dependent force to prepare all coherent states and the squeezed vacuum states Leibfried et al. (2003). For the trial state $D(\alpha=1)(|0\rangle-|1\rangle)$, we first prepare $|0\rangle-|1\rangle$, followed by a displacement of $\alpha=1$.

The real and positive coefficients $x_{j+1}$ of the data states $|d_{m}\rangle$, shown in Fig. 2b, are the square root of population distribution $p(j)$, where $p(j)=|\langle d_{m}|j\rangle|^{2}$. The population distribution is determined from a Fourier transform of the time evolution while driving the red-sideband transition

where $P_{e}$ is the probability to detect the ion in the state $|e\rangle$, $\Omega_{\textrm{rsb}}$ is the Rabi frequency of the red-sideband, and $\gamma$ is the decoherence rate of motional states. To determine $p(j)$ a fit constrained to be non-negative and $\sum_{j=0}^{6}p(j)=1$ is applied to Eq. 3 for $j=0$ to $6$.

Supplementary Material: Experimental quantum-enhanced machine learning over infinite dimensions