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Home , Quantum machine learning

Protocol for implementing nonparametric learning with trapped ions

Dan-Bo Zhang,1 Shi-Liang Zhu,1, 2, ∗ and Z. D. Wang3, 1, † 1Guangdong Provincial Key Laboratory of Quantum Engineering and Quantum Materials, GPETR Center for Quantum Precision Measurement, SPTE and SPTE and Frontier Research Institute for Physics South China Normal University, Guangzhou 510006, China 2National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China 3Department of Physics and Center of Theoretical and Computational Physics, The University of Hong Kong, Pokfulam Road, Hong Kong, China Nonparametric learning is able to make reliable predictions by extracting information from simi- larities between a new set of input data and all samples. Here we point out a quantum paradigm of nonparametric learning which offers an exponential speedup over the sample size. By encoding data into quantum feature space, similarity between the data is defined as an inner product of quantum states. A quantum training state is introduced to superpose all data of samples, encoding relevant information for learning in its bipartite entanglement spectrum. We demonstrate that a trained state for prediction can be obtained by entanglement spectrum transformation, using quantum matrix toolbox. We further work out a feasible protocol to implement the quantum nonparametric learning with trapped ions, and demonstrate the power of for .

Introduction.– Machine learning extracts useful infor- defined later [20], we show that relevant important in- mation from data for prediction. The extraction can formation for learning is represented by the bipartite en- be categorized into parametric and nonparametric learn- tanglement spectrum of |ψAi [21], and different kinds of ing [1,2]. Parametric learning distills knowledge of regression can be proposed by choosing different types data into parameters of a function, e.g., neural networks. of entanglement spectrum transformation. The transfor- However, the form of function may set a model bias or a mation involves for matrix inversion limitation. Without a predetermined form of a function, using auxiliary qumodes (continuous variables) [21, 22]. nonparametric learning can make predictions by extract- We further propose a feasible scheme to implement this ing information of similarities between new data and all quantum nonparametric learning with trapped ions [23– samples, with the appropriate sample weighting related 25], and demonstrate the power of quantum superpo- to correlation of samples. This can utilize a self-defined sition for machine learning. Our work provides a new kernel that may better capture the similarity between insight for machine learning by exploiting entanglement data, while on the other hand, it requires a large num- structure of quantum superposed training data. ber of samples and the runtime is polynomial with the Nonparametric regression.– Let us first introduce non- sample size, which is time-consuming for big data. parametric learning. Given a training dataset of M points {x(m), y(m)} (with m = 1, 2, ··· ,M), where In quantum setting, machine learning can be en- x(m) ∈ RN is a vector of N features and y(m) ∈ R is hanced with processing [3–13]. the target value, the goal is to learn an input-output While quantum algorithms of nonparametric learning function, which can be used to predicty ˜ for new data x˜. were studied for Gaussian processes [14–17], we focus A parametric regression is to find a function f(x), e.g., a on more general cases of nonparametric learning and its linear model, f(x) = wT x, parametrized by a matrix w. enhancement by exploiting quantum advantages. First, A nonparametric learning, instead, directly establishes a encoding classical data x into |ψxi can

arXiv:1906.03388v2 [quant-ph] 8 Jan 2020 prediction based on a weighted average over the similar- take advantages of quantum-enhanced feature spaces for ity between new data x˜ and each training data, namely, highly nonlinear feature map [12, 13, 18], which is de- sirable for complicated machine learning tasks. Second, all data of samples can be superposed, and querying M X (m) of similarities can be achieved in a quantum parallel y˜ = αmκ(x , x˜), (1) way. Moreover, correlations of data can be extracted and m=1 transformed more efficiently with quantum matrix tool- where κ(x(m), x˜) defines the similarity between data box [5,7, 19], including exponentiation and can be chosen beforehand. The weighting α = and matrix inversion. T (α1, ..., αM ) , for instance, can be determined by min- In this Letter, we illustrate a quantum paradigm for imizing the least-square loss function

nonparametric learning by elaborating on a regression M M task and its physical implementation. With a superpo- X (m) (m) 2 X 2 L(α) = (˜y − y ) + χ αm. (2) sition of all samples into a quantum training state |ψAi m=1 m=1 2

Here the χ-term is a L2 regularization term that makes a constraint on the weighting of each sample, which is necessary for avoiding over fitting. The combination of Eq. (1) and Eq. (2) is a kernel ridge regression. The solu- tion turns to be α = (K +χI)−1y, where K is the covari- (m1) (m2) ance matrix with elements Km1,m2 = κ(x , x ), and y = (y(1), ..., y(M))T . The prediction can be written T −1 (m) asy ˜ = y (K + χI) κ, where κm = κ(x , x˜). Nonparametric regression on a quantum computer can be reformulated to exploit quantum properties. First, classical data x is encoded into a quantum state |ψxi, which exploits the representation power of feature Hilbert space with highly nonlinear feature map [12, 13, 18].

The similarity between two data is defined as Km1,m2 = hψx(m1) |ψx(m2) i. Second, training and prediction can be performed on superposed quantum states of all training FIG. 1. Entanglement entropy vs the number of samples, data. To illustrate this idea, we take a superposition of where 40 random datasets are chosen for each set of samples. 1 The error bars denote standard deviations. The insert shows (m) − 2 P the training dataset {x } → M m |mi|ψx(m) i ≡ 1 that the mean-square-error decrease with the entropy. (m) − 2 P (m) |ψAi, {y } → |y| m y |mi ≡ |yi. The predic- tion is done by evaluating an overlapping between two states [20, 21]: the query state for a set of new data, Matrix inversion.– An efficient quantum algorithm y ⊗ x˜ → |ψ i ≡ |yi|ψ i, and a trained state |ψ + i that R x˜ A can be developed to obtain |ψA+ i from |ψAi. Note evolves from |ψAi, i.e., that the covariance matrix can be evaluated as ρK = K/TrK = Tr1|ψAihψA|(partial trace of the addressing y˜ = hψR|ψA+ i. (3) 2 registers |mi) and I ⊗ ρK |ψAi = λi |ψAi. The required evolution is given by A derivation is shown in Supplemental Material (SM) [26]. Eq.(3) represents a quantum version of non- −1 |ψA+ i = I ⊗ B |ψAi. (4) parametric learning, serving as a generalization of quan- tum linear regression in Ref. [20, 21] to nonlinear cases. where B = ρK + χI. Therein, learning is manifested in a proper trained The non-unitary operator B−1 is a matrix inver- state |ψA+ i. A naive choice of |ψA+ i = |ψAi means sion and its quantum algorithm can exhibit exponential all training data has equal weighting, neglecting correla- speed-up. We take an approach for the matrix inver- tions between the training data. A wisdom from quan- sion of B by writing it into a combination of unitary tum information is to investigate entanglement struc- −1 R ∞ operators [30, 31]. Inspired by b = −∞ dxδ(bx) = ture of the bipartite state |ψAi. Correlations between R ∞ dxdy exp(ibxy), we consider B|bi = b|bi, we have data reflect in a Schmidt decomposition of the train- −∞ ing state, |ψ i = P λ |u i|v i. For a least-square loss Z ∞ A i i i i −1 P B = dqxdqy exp(iBqxqy) in Eq. (2), the trained state |ψA+ i = c g(λi)|uii|vii, i −∞ where g(λ) = λ [21] (see SM [26]). The transforma- λ2+χ ∝ h0 |h0 | exp(iBqˆ qˆ )|0 i|0 i, (5) tion of Schmidt coefficients λ → g(λ) can be considered px py x y px py as entanglement spectrum transformation [27], and dif- where |0ip is zero momentum eigenstate. It can be seen ferent choices of g(λ) may correspond to different types that B−1 can be written as an average of unitary operator of regression [28]. exp(iBqˆxqˆy) over the infinite squeezing state |0px i|0py i of It is inspiring to investigate the role of entanglement momentums px and py. entropy S of bipartite quantum state |ψAi for machine The state transformation |ψAi → |ψA+ i can be learning. For illustration, we use squeezing-state en- implemented as follows: B−1 performs on the initial coding with a varied squeezing factor s for the Boston state |ψAi|0px i|0py i, and then project two qumodes dataset [29]. The similarity function between two sam- −1 onto |0px i|0py i. To implement B , we can write −s2|x(m1)−x(m2)|2 ples is Km1,m2 = e , and samples are less exp(iBqˆxqˆy) = exp(iρK qˆxqˆy) exp(iχqˆxqˆy). The first part distinguishable for smaller s. As seen from Fig. (1), S in- exp(iρK qˆxqˆy) can be generated by density matrix expo- creases with the number of samples and saturates faster nentiation by sampling from multiple copies of quantum for smaller s. Moreover, the mean-square error decreases state |ψAi [7, 32]. The second part is just a basic two- with S, indicating that the entanglement entropy may be qumode gate. related to the model capacity that quantifies the ability Quantum algorithm.– We now turn to work out a quan- to fit complicated data (see SM [26]). tum algorithm for nonparametric regression, basically 3

p = 1 (1 + |hψ0 |ψ i|2) is used to infer the prediction √2 A+ R y˜ ∝ 2p − 1, up to a sign. Quantum advantages.– We now elaborate that the above algorithm has an exponential speed-up. Using quantum random access memory |ψAi can be prepared in a runtime of O(log M). It takes O(ε−1) copies −1 of |ψAi, thus a runtime of O(ε log M) to perform exp(iρK qˆxqˆy)[21, 22], for a desired accuracy ε. The success rate of homodyne detection is O(s−4) and this procedure thus requires O(s4) (see SM [26]). In total, the runtime scales as O(s4ε−1 log M). The exponential speed-up relies on the capacity of superposition. If ran- domly chosen M 0 < M training data is superposed for each copy [34], then the number of copies should be in- FIG. 2. Illustration of the quantum algorithm. (a). Matrix M creased O( M 0 ) times. To retain exponential speed-up inversion algorithm for a matrix B = ρK + χI that transform requires M 0/M ∼ O(1). −1 |ψAi into |ψA+ i = B |ψAi, using two auxiliary qumodes that are post-selected into zero momentum. (b). A swap Another potential quantum advantage comes from + quantum feature map when encoding x into |φ i. Re- test that evaluates the inner product between |A i and |ψRi, x which can be used to infer the prediction for input data a˜. markably, continuous variable provides infinite dimen- sion Hilbert space with highly nonlinear feature maps. For instance, encoding into a Gaussian state, such as following techniques in Ref. [21]. The main steps are |φxi = ⊗i|xiic (|xiic denotes a with show in Fig. 2, where steps 1 − 4 illustrated in Fig. 2a a displacement xi), corresponds to a Gaussian kernel, 2 transform |ψ i to |ψ + i, and step 5 illustrated in Fig. 2b −|u−v| /2 A A since hφu|φvi = e . Classically intractable in- implements the prediction. stantaneous quantum polynomial or continuous vari- 1. State preparation. Prepare the data state |ψAi, able instantaneous quantum polynomial circuits are pur- the query state |ψRi, and a two-qumode state |sipx |sipy , sued [35, 36]. Moreover, a promising direction is to find 1/2 −1/4 R −s2p2/2 where |sip = s π dpe |pip. |ψAi can be encoding schemes that can better represent similarities prepared efficiently with a quantum random access mem- between data for specified tasks, and thus require less P ory [33]. It uses the addressing state m |mi to access training data and better generalization, such as predict- the memory cells storing quantum states |φx(m) i in train- ing energies for molecules [37, 38]. ing data registers. Also, two qumodes are initialed in a Quantum operations required in trapped ions.– Imple-

finite squeezing state |sipx |sipy . menting the above quantum algorithm requires hybrid 2. Quantum phase estimation. Perform U = discrete and continuous variable . exp(iρK qˆxqˆy) on |ψAi|sipx |sipy , where U is constructed Some promising candidates for quantum computation, with the density matrix exponentiation method [7, 22, such as superconducting in a circuit-QED and 32]. The quantum state becomes trapped ions, have this property. Here we take trapped

2 2 2 ions as the platform [23–25] to illustrate the details. We Z −(qx+qy )/2s X λie iλ2q q dq dq |u i|v i|q i |q i e i x y consider trapped ions in a Paul trap, and take L internal x y s i i x qx y qy i levels of each ions as a qudit to encode the discrete vari- (6) ables and local transverse phonon modes (along x and y directions) [39, 40] to encode the continuous variables, iχqˆ qˆ 3. Regularization. Perform e x y on two qumodes. while the longitudinal collective modes along z direction Here χ is a preset hyperparameter. The state is the same serves as the bus modes to connect any two ions. No- i(λ2+χ)q q as Eq.(6) by changing the phase factor to e i x y . tably both internal states and phonon modes are well 4. Singular-value transformation. Project two controllable in trapped ions [40–47]. qumodes into the squeezing state |si |si , and the state px py We outline quantum operations required for the pro- 0 P turns to be |ψA+ i = i f(λi, s, χ)|uii|vii, approximating posed algorithm (see SM [26]). We first address the op- λi |ψA+ i, where f(λi, s, χ) = q . 4 +(λ2+χ)2 erations acting on single ion, denoting as the j-th ion. s4 i iθσjn 5. Prediction. For new data x˜, the predictiony ˜ ∝ A single gate R(θ, n) = e acting on any two 0 internal levels of the j-th ion with high fidelity is real- hψR|ψA+ i can be accessed with a . After the conditional swap operation, an entangled state is ob- izable, where σjn is a Pauli matrix along the direction tained, |Ψi = √1 (|0i ⊗ |ψ0 i|Ψ i + |1i ⊗ |Ψ i|ψ0 i. n. Operations on a motional mode include Pα(θ) = A+ R R A+ † ∗ † 2 iθa ajα hajα−h a Then, a Hadamard gate is performed on the qubit, fol- e jα , displacement operator Dα(h) = e jα − ln s (a2 −a†2 ) lowed with a projection into |0i, whose success rate and squeezing operator Sα(s) = e 2 jα jα with 4

† α = x, y [23, 44, 48–50], where ajα (ajα) is the cre- j = b, c−ions, can be generated with Dirac type op- ate (annihilation) operator of the α phonon mode. A erations (see SM [26]). Also two qumodes of the c- x y iχqˆj qˆj controlled phase gate Cq = e coupling both mo- ion are prepared in a squeezing state |sipx |sipy . For tional modes can be realized by manipulating the trap the quantum phase estimation, the unitary operation potential. By using red and blue side excitations in- U = exp(iρK qˆxqˆy) is constructed with the density ma- duced by lasers, internal and motional states can be cou- trix exponentiation method [7, 21, 22, 32], x x pled, e.g., obtaining Dirac type operators H1 = gqˆj σj iδtqˆxqˆy Scv 0 −iδtqˆxqˆy Scv y y Trρ(e ρ ⊗ ρ e ) and H2 = gqˆj σj [42, 43]. Then the hybrid opera- iησ qˆxqˆy iρδtqˆxqˆy 0 iρδtqˆxqˆy 2 tor W(η) = e jz j j , which is important for quan- = e ρ e + O(δt ). (7) tum phase estimation, can be constructed by repeat- Here ρ = ρ is a mixed state encoded in the mo- edly applying 1/(g2δt2) times of the quantum evolution K iH δt iH δt −iH δt −iH δt −[H ,H ]δt2 3 tional states of c-ion (the internal states are traced out), e 2 e 1 e 2 e 1 = e 1 2 + O(δt ). 0 and ρ ≡ |ψAihψA| is a state on b-ion. The con- As for two ions, besides the standard controlled- ditional swap operator eiδtqˆxqˆy Scv is constructed from NOT gate [41], a beam-splitter defined as B(θ) = CS HaWa(δt)HaCS [38], where CS swaps motional iθ(a† a +a† a ) cv cv cv e jα j+1α j+1α jα is needed, and it was theoretically states of b-ion and c-ion, conditioned on the qubit state of proposed [40] and then experimentally achieved recently a-ion initialized in |+i state. The one-qubit-two-qumodes [51]. These two operators thus couple qubit states or coupling Wa(δt) performs on the a-ion, and Ha is a qumodes from different ions. Furthermore, a coupling Hadamard gate acting on the a-ion. Multiple copies of of one qubit from an ion and a qumode from another c-ion are required and each is encoded with mixed state ion is possible with Dirac type Hamiltonians where ρ in the internal states. Conditional swap operations are and momentum (position) come from different ions. Nec- sequentially performed on b-ion and a new c-ion and swap essary quantum operations on three ions includes con- their motional states, effectively giving a U operation on trolled swap operators, for which one ion provides a qubit b-ion. to control a swap for other two ions, either on internal After applying U on the b-ion, a regularization can be iχqˆxqˆy states or motional states. The former has been real- realized by applying Cq = e on the two motional ized experimentally in trapped ions [52]. On the other modes of c-ion. A measurement projects two qumodes hand, precision measurement can be implemented for of the c-ion onto |sipx |sipy . The b-ion is on target state both qubits [23] and qumodes [53]. Those unitary oper- † |ψA+ i. After an evolution U , where UR|0i = |ψRi, a ators and measurements serve as building blocks for the R projective measure on |gi|0ic with the success probability quantum algorithm of nonparametric regression as well 2 p = |hψR|ψA+ i| can infer the prediction for new datax ˜. as other hybrid quantum information processing tasks. This implementation scheme can demonstrate a re- Physical implementation with trapped ions.– We illus- markable quantum-enhanced property. The above trate the implementation with a simple example. We density matrix exponentiation can use partial train- just take one ion to encode the training dataset, that is, ing dataset for each time [34], e.g., use ρ ∼ using only one ion to represent one copy of state |ψAi. P (m) (m) |φj(x )ihφj(x )|, where RM represents to To this end, we choose L(= M) internal levels of the ion m∈RM randomly choose RM samples in the training dataset, as a qudit to encode the M points dataset and two local and we thus choose L = RM internal levels to represent transverse phonon modes (along x and y directions) to state |ψAi. Therefore, in the experiments, we can com- encode the continuous variables (N=2). pare the results of RM = 1, 2, ··· ,M randomly chosen The implementation needs four types of ions which we data from the dataset for each copy. We calculate the pre- denote as a, b, c, d− ions. 1). An a-ion provides a qubit diction errors as a function of the number of the c-ions, and two qumodes as auxiliary modes. 2). A b-ion is and the results are shown in Fig. 3b. Under the con- used to store the state |ψAi that encodes all data. L dition of same accuracy, the number of c-ions increases internal levels and two local motional modes along the with the decrease of RM ; similarly, the prediction errors x, y directions are used. On this ion the state will be decrease for a large RM . Therefore, it is a clear evidence transformed into the target state |ψA+ i. 3). Several to demonstrate the power of superposition for quantum c-ions, the number of which depends on the accuracy nonparametric learning. A remarkable result presented required for the algorithm, are used for constructing the here is that, a Paul trap with around ten ions, which has unitary operator U on the b-ion. Each c-ion is initialized been realized in several groups [54–57], can demonstrate in the state |ψAi. 4). A d-ion encodes input data for the quantum-enhanced property for prediction into quantum state |ψRi. learning. (As for the scheme scalability, it is discussed The scheme for nonlinear regression is schematically in SM [26].) shown in Fig. 3a. In the state preparation, the gen- To conclude, we have illustrated a quantum paradigm PM eralized Schrodinger cat states |ψjAi = m=0 |mji ⊗ of nonparametric learning that can fully exploit quantum (m) (m) (m) (m) |φj(x )i, where |φj(x )i = |xjx i|xjy i for both advantages with realistic physical implementation. The 5

FIG. 3. (a). A quantum procedure that transforms |ψAi into |ψA+ i on b-ion, assisted by an a-ion providing a qubit for control and two qumodes for matrix inversion, and many c-ions initialized in |ψAi serving as quantum software states for quantum phase estimation. Note the swap only performs on motional states (red dash lines). (b). Density matrix exponentiation where partial training dataset is superposed. Here RM = 1, 2, 3, 4 stands for the number of randomly chosen samples for each copy. The region with dashed lines represents accessible zone for a trade-off between error and the number of ions involved, constraint by the maximum available c-ion Nt = 20. above-proposed experimental scheme has paved the way [9] V. Dunjko, J. M. Taylor, and H. J. Briegel, “Quantum- for quantum machine learning. enhanced machine learning,” Phys. Rev. Lett. 117, This work was supported by the National Key Re- 130501 (2016). [10] S. Lloyd, S. Garnerone, and P. Zanardi, “Quantum al- search and Development Program of China (Grant No. gorithms for topological and geometric analysis of data,” 2016YFA0301800), the National National Science Foun- Nature Communications 7, 10138 (2016). dation of China (Grants No. 91636218, No.11474153,and [11] Seth Lloyd and Christian Weedbrook, “Quantum gener- No. U1801661), the Key R&D Program of Guangdong ative adversarial learning,” Phys. Rev. Lett. 121, 040502 province (Grant No. 2019B030330001), and the Key (2018). Project of Science and Technology of Guangzhou (Grant [12] Vojtch Havlek, Antonio D. Crcoles, Kristan Temme, No. 201804020055). Aram W. Harrow, Abhinav Kandala, Jerry M. Chow, and Jay M. Gambetta, “Supervised learning with quantum-enhanced feature spaces,” Nature 567, 209–212 (2019). [13] Maria Schuld and Nathan Killoran, “Quantum machine learning in feature hilbert spaces,” Phys. Rev. Lett. 122, ∗ [email protected] 040504 (2019). † [email protected] [14] Siddhartha Das, George Siopsis, and Christian Weed- [1] C. M. Bishop, and machine learning, brook, “Continuous-variable quantum gaussian process Vol. 1 (Springer, 2006). regression and quantum singular value decomposition of [2] Trevor Hastie, Robert Tibshirani, and Jerome Friedman, nonsparse low-rank matrices,” Phys. Rev. A 97, 022315 The Elements of Statistical Learning, 2nd ed. (Springer, (2018). 2009). [15] Zhikuan Zhao, Jack K. Fitzsimons, and Joseph F. Fitzsi- [3] J. Biamonte, P. Wittek, N. Pancotti, P. Rebentrost, mons, “Quantum-assisted gaussian process regression,” N. Wiebe, and S. Lloyd, “Quantum machine learning,” Phys. Rev. A 99, 052331 (2019). Nature 549, 195–202 (2017). [16] Zhikuan Zhao, Alejandro Pozas-Kerstjens, Patrick [4] Sankar Das Sarma, Dong-Ling Deng, and Lu-Ming Rebentrost, and Peter Wittek, “Bayesian Duan, “Machine learning meets quantum physics,” Phys. on a quantum computer,” Quantum Machine Intelligence Today 72, 48–54 (2019). 1, 41–51 (2019). [5] A. W. Harrow, A. Hassidim, and S. Lloyd, “Quantum [17] Zhikuan Zhao, Jack K. Fitzsimons, Michael A. Osborne, algorithm for linear systems of equations,” Phys. Rev. Stephen J. Roberts, and Joseph F. Fitzsimons, “Quan- Lett. 103, 150502 (2009). tum algorithms for training gaussian processes,” Phys. [6] N. Wiebe, D. Braun, and S. Lloyd, “Quantum algorithm Rev. A 100, 012304 (2019). for data fitting,” Phys. Rev. Lett. 109, 050505 (2012). [18] K. Mitarai, M. Negoro, M. Kitagawa, and K. Fujii, [7] S. Lloyd, M. Mohseni, and P. Rebentrost, “Quantum “ learning,” Phys. Rev. A 98, 032309 principal component analysis,” Nat. Phys. 10, 631–633 (2018). (2014). [19] Andrs Gilyn, Yuan Su, Guang Hao Low, and Nathan [8] P. Rebentrost, M. Mohseni, and S. Lloyd, “Quantum Wiebe, “Quantum singular value transformation and be- support vector machine for big data classification,” Phys. yond: exponential improvements for quantum matrix Rev. Lett. 113, 130503 (2014). 6

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