Quantum Machine Learning over Infinite Dimensions

Abstract

Machine learning is a fascinating and exciting field within computer science. Recently, this excitement has been transferred to the quantum information realm. Currently, all proposals for the quantum version of machine learning utilize the finite-dimensional substrate of discrete variables. Here we generalize quantum machine learning to the more complex, but still remarkably practical, infinite-dimensional systems. We present the critical subroutines of quantum machine learning algorithms for an all-photonic continuous-variable quantum computer that can lead to exponential speed-ups in situations where classical algorithms scale polynomially. Finally, we also map out an experimental implementation which can be used as a blueprint for future photonic demonstrations.Introduction -We are now in the age of big data [1]. An unprecedented era in history where the storing, managing and manipulation of information is no longer effective using previously techniques. To compensate for this, one important approach in manipulating such large data sets and extracting worthwhile inferences, is by utilizing machine learning techniques. Machine learning [2,3] involves using specially tailored 'learning algorithms' to make important predictions in fields as varied as finance, business, fraud detection, and counter terrorism. Tasks in machine learning can involve either supervised or unsupervised learning and can solve such problems as pattern and speech recognition, classification, and clustering. Interestingly enough, the overwhelming rush of big data in the last decade has also been responsible for the recent advances in the closely related field of artificial intelligence [4].Another important field in information processing which has also seen a significant increase in interest in the last decade is that of quantum computing [5]. Quantum computers are expected to be able to perform certain computations much faster than any classical computer. In fact, quantum algorithms have been developed which are exponentially faster than their classical counterparts [6,7]. Recently, a new subfield within quantum information has emerged combining ideas from quantum computing with artificial intelligence to form quantum machine learning [8].These discrete-variable schemes have observed a performance that scales logarithmically in the vector dimension, such as supervised and unsupervised learning [9], support vector machine [10], cluster assignment [11] and others [12][13][14][15][16][17][18]. Initial proof-of-principle experimental demonstrations have also been performed [19][20][21][22]. It was mentioned in [23], that certain caveats apply to quantum machine learning. However, since then these caveats (relating to sparsity, condition number, epsilon precision, quantum output), have been closed or applications found where they are not a concern, cf. [8,10,18,24].

Complexity measures and uncertainty relations of the high-dimensional harmonic and hydrogenic systems

“…The Fisher information, contrary to the Rényi, Shannon and Tsallis entropies, is a local measure of spreading of the density ρ( r) because it is a gradient functional of ρ( r), so that it is very sensitive to the density fluctuations. Moreover, these entropic measures allow us to characterize and identify some relevant quantum phenomena such as e.g., quantum phase transitions [55], fractality [56], machine learning [57] and the spectral avoided crossings in atoms and molecules [58,59]. To a great extent this is because the entropic measures satisfy various relevant mathematical properties [60][61][62] and the position-momentum uncertainty relations [51,63,64] (see also [32,42,48]) given by…”

Universal Quantum Computing with Arbitrary Continuous-Variable Encoding

“…Arbitrary rotation on the Y -Z plane of the D-logical Bloch sphere is accomplished by a two-mode exponential-swap (E-swap) operation [20,21], i.e., exp(iθŜ 2n−1,2n ) = exp(iθX Dn ) .…”

Universal continuous-variable quantum computation without cooling

“…(2), while the implementation of exponential-swap can be found in Refs. [40,45,46]. We first note that the single-mode and two-mode exponential-parity gates can be expressed as e iθP = cos θÎ + i sin θP , e iθP⊗P = cos θÎ + i sin θP ⊗P Consider the auxiliary qubit is prepared as |+ and a single-qumode state |ψ .…”

“…Alternatively, the logic gates can be implemented by applying beamsplitter and phase-shift operations, together with the second order hybrid interaction of which the Hamiltonian is H 2 ∝Ẑ Aâ †â , whereẐ A is the Pauli Z operator of an auxiliary qubit A (See Appendix II and Refs. [40,45,46]),…”

“…Alternatively, the logic gates can be implemented by applying beamsplitter and phase-shift operations, together with the second order hybrid interaction of which the Hamiltonian is H 2 ∝Ẑ Aâ †â , whereẐ A is the Pauli Z operator of an auxiliary qubit A (See Appendix II and Refs. [40,45,46]),|ψ L k L l is the four-qumode physical state of the kth and lth logical qubits.is the controlled-parity operator acting on the ith qumode;) is the 50:50 beam splitter between the two qumodes of the kth logical qubit. The auxiliary qubit state is the same as |+ A = (|0 A +|1 A )/ √ 2 before and after all operations;R σ (θ) ≡ exp(iθσ A ) is the rotation of the auxiliary qubit along the Pauli σ axis.…”