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An introduction to quantum machine learning: from quantum logic to quantum deep learning

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Abstract

The aim of this work is to give an introduction for a non-practical reader to the growing field of quantum machine learning, which is a recent discipline that combines the research areas of machine learning and quantum computing. This work presents the most notable scientific literature about quantum machine learning, starting from the basics of quantum logic to some specific elements and algorithms of quantum computing (such as QRAM, Grover and HHL), in order to allow a better understanding of latest quantum machine learning techniques. The main aspects of quantum machine learning are then covered, with detailed descriptions of some notable algorithms, such as quantum natural gradient and quantum support vector machines, up to the most recent quantum deep learning techniques, such as quantum neural networks.

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Notes

  1. The C01 can be referred in literature as \(\overline {\mathrm {C}}\)NOT, and its control value should be |0〉 in order to change.

  2. Kullback–Leibler divergence or KL divergence is a measure of how far away two probability distributions are from each other.

  3. Fubini-Study metric, seen as the Fisher information matrix in quantum field.

  4. See https://www.statista.com/statistics/993634/quantum-computers-by-number-of-qubits/ for a general idea of the trend.

  5. For a better comprehension of the phenomenon, see Feynman (1965)

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Appendix

Appendix

1.1 A quantum mechanics: basic elements

In this section, some basic concepts of quantum mechanics are introduced. Olivares (2020)

Since quantum systems often behave very differently from classical systems, a novel representation is needed in their description. Understanding the meaning of this representation and its mechanisms allows the reader to comprehend the basics of quantum computation and quantum machine learning (QML).

1.2 A.1 System, states, measures

With the term physical system, we refer to any portion of the physical universe that can be described and analyzed. Classical systems are described by the laws of classical physics, an object falling from a cliff, an electric charge in space, etc.

Quantum systems, on the other hand, are described by the laws of quantum physics. Ensemble of atoms or particles are typical example of quantum systems. Actually, any event happening at atomic or subatomic scale can be described in term of a quantum system. Now, it is known that classical physical systems have a deterministic nature: given an initial state, the system will produce always the same output. For example, the behaviour of a moving object can be predicted given a certain set of initial parameters. More precisely, the result of any measurement on that object (position at a certain time, velocity, rotational speed) can be predicted in advance. On the other hand, quantum systems have an intrinsic stochastic behaviour. One cannot predict with certainty the result of a measure, but only its probability. In other words, a certain quantum measurement on a given system can have multiple outcomes, and each of these outcomes will manifest itself with a given probability, which can be predicted in advance. To represent this situation, the concept of quantum state is introduced. A state is an object that allows for the calculation of events probabilities, and contains all the information about a given system. To refer to a state, the Dirac’s notation is used: a state of the system is associated with an object called ket, through the notation |x〉.

The simplest possible quantum system is called qubit, that has the following representation:

$$ |{\phi}\rangle = a_{+} |{+}\rangle + a_{-} |{-}\rangle $$
(77)

where a+ and a are complex numbers. When a measurement is realized on this system, it can only have one of two possible outcomes, which are represented by |+〉 and |−〉. Each of these outcomes has a probability of respectively |a+|2 and |a|2. It is said that the system is a superposition of two states. The two states “coexists” inside the system, and one or the other can be revealed with a measurement. Mathematically, the state |ϕ〉 is a vector in Hilbert Space \({\mathscr{H}}\), of which |+〉 and |−〉 are a basis. A practical example of this situation might be the following: the state |ϕ〉 represent the position of a single, static particle. This particle can either be in two (fixed) position. As said, a quantum state (a ket) represent a physical system. In the case reported in Eq. 77, the state |+〉 represents the particle in the first position, while the state |−〉 represents the particle in the second position. The state |ϕ〉, finally, represents the particle whose position is unknown. The particle is both in the first and second position, with given probabilities.

It is important to remark that the state |ϕ〉 does not simply mean since the position is unknown, it can be one or the other, but physically represent a particle that is both in position one and two Footnote 5.

By implicitly assuming that the only possible outcomes of the measure are |+〉 and |−〉, it is possible to write:

$$ |{\phi}\rangle = a_{+} |{+}\rangle + a_{-} |{-}\rangle \longrightarrow \left( \begin{array}{cc} a_{+} \\ a_{-} \end{array}\right) $$
(78)

In other words, the state is represented as a linear combination of the basis vectors of the Hilbert space \({\mathscr{H}}\).

$$ |{\phi}\rangle = a_{+}\left( \begin{array}{cc} 1 \\ 0 \end{array}\right) + a_{-} \left( \begin{array}{cc} 0 \\ 1 \end{array}\right) $$
(79)

where:

$$ \begin{array}{@{}rcl@{}} |{+}\rangle &\longrightarrow \left( \begin{array}{cc} 1 \\ 0 \end{array}\right)\\ |{-}\rangle &\longrightarrow \left( \begin{array}{cc} 0 \\ 1 \end{array}\right) \end{array} $$

Equivalently, another representation for states is defined: the bra.

$$ \langle{x}| = |{x}\rangle^{*} $$
(80)

where the symbol (∗) denotes the adjointness operation. Given |ϕ〉 in Eq. 77:

$$ \langle{\phi}| = a^{*}_{+} \langle{+}| + a^{*}_{-} \langle{-}| \longrightarrow \left( \begin{array}{cc} a^{*}_{+} & a^{*}_{-} \end{array}\right) $$
(81)

It is then clear that the product between bra and ket is equivalent to the inner product between vectors:

$$ \langle{\phi}| = a^{2}_{+} + a^{2}_{-} $$
(82)

While the ket-bra product yields a matrix:

$$ \langle{\phi}|{\phi}\rangle = \left( \begin{array}{cc} a^{2}+ & a_{+}a^{*}_{-} \\ a_{-}a^{*}_{+} & a^{2}_{-} \end{array}\right) $$
(83)

1.3 A.2 Operators

An operator is an object that, when applied to a quantum state, yields another quantum state:

$$ A |{\phi}\rangle = |{\phi^{\prime}}\rangle $$
(84)

For a two-level system, it’s possible to write:

$$ A |{\phi}\rangle = A(a_{+} |{+}\rangle + a_{-} |{-}\rangle) = a_{+}A|{+}\rangle + a_{-} A |{-}\rangle $$
(85)

So, in this case, an operator can be thought as a square matrix. In quantum mechanics, every observable (speed, momentum, position, etc.) is an operator. It it possible to obtain the mean value of an observable O on a given state|ϕ〉 with the following expression:

$$ \langle{O}\rangle = \langle{\phi}|{O}|{\phi}\rangle $$
(86)

It is important to remark that any information on a given quantum state must be obtained through a measurement. For this reason, in quantum computing and quantum machine learning the concepts of observable and measurement are very important, since they represent the only way to interact with a quantum system.

Another important concept in quantum mechanics is the Wave Function collapse: basically, any form of measurements on a quantum state affects it irreversibly, causing the collapse of the function. For example, supposing that a measurement on a two-level quantum state as in Eq. 77 is made. The result of the measurement is |+〉. After the measurement, the state of the system will always be |+〉, as if the other component |−〉 ceased to exist. As stated by the Copenhagen interpretation of quantum mechanics: when a measurement is done on the system, its state changes, collapsing in the state that resulted by that measure. Any measurement, in conclusion, affects the system irreversibly.

1.4 A.3 Density matrix

An alternative way to represent a quantum state is through its density matrix, defined as:

$$ \rho_{\phi} = \langle{\phi}|{\phi}\rangle $$
(87)

This form of representation is particularly useful, since, given an observable O, its mean value on a state |ϕ〉 can be written as:

$$ \langle{O}\rangle = \text{Tr}(O\rho_{\phi}) $$
(88)

1.5 A.4 Pauli matrices

For a two level system, both operators and density matrix are 2 × 2 matrices. It is common to represent any 2 × 2 matrix as a linear combination of the following matrices:

$$ \begin{array}{@{}rcl@{}} I = \left( \begin{array}{cc} 1\\& 1 \end{array}\right) \sigma_{1} = \left( \begin{array}{cc} 0&1\\1&0 \end{array}\right)\\ \sigma_{2} = \left( \begin{array}{ll} 0-i\\i\ 0 \end{array}\right) \sigma_{3} = \left( \begin{array}{cc} 0&-1\\1&0 \end{array}\right) \\ \end{array} $$

where σn are called Pauli matrices. In other words, on a two level system, every observable and every density matrix (and so every state) can be represented as a linear combination of the Pauli Matrices and the identity. These operators, in other words, constitute a basis of the space of observables and states for two level systems, and will be very useful in the next sections.

1.6 A.5 Hamiltonians and temporal evolution

Until now, when describing a quantum system, time dependency has not been accounted for. In other words, only static system have been considered. In quantum mechanics, quantum systems are usually subject to the action of time: they evolve. It is possible, then, to write:

$$ |{\phi(t)}\rangle $$
(89)

where the parameter t makes the time dependency explicit. As mentioned earlier, in order to modify a quantum state, an operator is needed. This holds also for time dependency. The temporal evolution operator is defined as:

$$ S(t, t_{0})|{\phi(t_{0})}\rangle = |{\phi(t)}\rangle $$
(90)

It is also possible to show that this operator can be written, for infinitesimal time variations as:

$$ S(t_{0} + \varepsilon, t_{0}) = I - i\varepsilon \hat{H}(t_{0}) $$
(91)

In other words, the operator S(t0 + ε,t0) shifts the state from time t0 to time t0 + ε.

The operator \(\hat {H}(t)\) is the hamiltonian of the system. While the utility of such expression might not be clear at first sight, it is important to remark that hamiltonians are central concepts of mechanical physics in general (Landau and Lifsits 2010). Also, they are very important in quantum computing, inside the so-called adiabatic model (Olivares 2020). It is possible to think of the hamiltonian as that operator which completely determines the temporal evolution of the system.

1.7 A.6 Entanglement of two qubit-states

It is possible to define a two qubit state from two single qubits, namely:

$$ \begin{array}{@{}rcl@{}} |{\phi_{A}}\rangle &=& \alpha_{A} |{+_{A}}\rangle + {\upbeta}_{A} |{-_{A}}\rangle \end{array} $$
(92)
$$ \begin{array}{@{}rcl@{}} |{\phi_{B}}\rangle &=& \alpha_{B} |{+_{B}}\rangle + {\upbeta}_{B} |{-_{B}}\rangle \end{array} $$
(93)

Then, the two qubit state can be defined as:

$$ |{\phi_{A} \phi_{B}}\rangle = |{\phi_{A}}\rangle|{\phi_{B}}\rangle = |{\phi_{A}}\rangle\oplus|{\phi_{B}}| $$
(94)

This is a first example of a separable state, i.e. a state that can be written as the tensor product between two different states. This is not always the case, for example:

$$ |{\phi_{AB}}\rangle = \frac{|{-_{A}}\rangle|{-_{B}}\rangle + |{+_{A}}\rangle|{+_{B}}\rangle} {\sqrt{2}} $$
(95)

This state is not separable: the two subsystems are entangled.

Quantum entanglement is at the heart of quantum mechanics, and gives rise to some counter-intuitive phenomena that can be exploited in the fields of quantum information and computation. Supposing, for example, that A and B represent two different particles in the entangled state, Eq. 95; one could imagine to perform a measurement on particle A, which gives result |−A〉. From that moment on, any measurement on particle B will yield the result |−B〉, because the state has “collapsed” in the first term of Eq. 95. The two subsystems are entangled in the sense that a measurement on one of the two affects both irreversibly. In other words, measurement on the two subsystems are perfectly correlated. The utilization of this phenomenon in quantum computation is a very active area of research and development.

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Alchieri, L., Badalotti, D., Bonardi, P. et al. An introduction to quantum machine learning: from quantum logic to quantum deep learning. Quantum Mach. Intell. 3, 28 (2021). https://doi.org/10.1007/s42484-021-00056-8

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