2. Quantum neural computation and quantum mechanics

In order to address quantum neural computation, we need to first introduce some notation, which is commonly used in quantum computation, namely, we use the standard Dirac's bra-ket notation, where a ket vector corresponds to a column vector and the bra vector is its conjugate transpose. Defining, then, the fundamental ket vectors 0j i and 1j i, respectively, as:

$$|\mathbf{0}\rangle = \begin{pmatrix} \mathbf{1} \\ \mathbf{0} \end{pmatrix}, |\mathbf{1}\rangle = \begin{pmatrix} \mathbf{0} \\ \mathbf{1} \end{pmatrix} \tag{1}$$

with the corresponding bra vectors 0h j and 1h j being defined, respectively, as the conjugate transpose of 0j i and 1j i, then, we can represent Pauli's operators as:

$$
\hat{\sigma}\_1 = |\mathbf{0}\rangle\langle\mathbf{1}| + |\mathbf{1}\rangle\langle\mathbf{0}| = \begin{pmatrix} \mathbf{0} & \mathbf{1} \\ \mathbf{1} & \mathbf{0} \end{pmatrix} \tag{2}
$$

$$
\hat{\sigma}\_2 = -i|\mathbf{0}\rangle\langle\mathbf{1}| + i|\mathbf{1}\rangle\langle\mathbf{0}| = \begin{pmatrix} \mathbf{0} & -i \\ i & \mathbf{0} \end{pmatrix} \tag{3}
$$

$$
\hat{\sigma}\_3 = |\mathbf{0}\rangle\langle\mathbf{0}| - |\mathbf{1}\rangle\langle\mathbf{1}| = \begin{pmatrix} \mathbf{1} & \mathbf{0} \\ \mathbf{0} & -\mathbf{1} \end{pmatrix} \tag{4}
$$

devices (Section 3.1); afterward (Section 3.2), we turn to the experimental implementation of quantum robotics and quantum decision with a more complex form of quantum neural computation in the form of a variant of quantum neural reinforcement learning (QNRL), applied to a problem of decision under risk, where the agent must learn the optimal action that leads to the highest expected reward in a

Artificial Intelligence - Applications in Medicine and Biology

The problem is first addressed in terms of the fundamental equations which employ quantum adaptive computation, namely quantum adaptive gates; then, we implement it experimentally on IBM's quantum computers and, afterward, we address the main Python code that was used to run the algorithm on these computers, thus, introducing quantum object-oriented programming (QOOP) and reflecting on its relevance for research on quantum artificial intelligence. While, in Section 3.1, the main goal is to illustrate the implementation of QUANNs in a case where QUANNs exhibit a greater efficiency over classical ANNs, in Section 3.2, our main goal is not to address the speed-up of quantum algorithms over classical ones or even the greater efficiency of quantum algorithms over classical ones, but rather to provide for a reflection on the first steps for a possible future where quantum computation is incorporated in different (classical) robotic systems by way of the internet of things and cloud-based access to quantum devices, and the role that quantum adaptive computation may play in such a future. In particular, in Section 3.2, we illustrate how a QUANN can become adaptive with respect to a problem that is given to it, in this case, a decision problem under risk, therefore, allowing us to address how QOOP can be employed to simulate an artificial agent, with a QUANN as its cognitive architecture, that must make a decision when presented a problem of classical decision under risk; therefore, our main goal in Section 3.2, from a computer science standpoint, is to address how a quantum artificially intelligent system decides when faced with a classical decision

In Section 4, we conclude with a chapter review and a reflection on future

In order to address quantum neural computation, we need to first introduce some notation, which is commonly used in quantum computation, namely, we use the standard Dirac's bra-ket notation, where a ket vector corresponds to a column vector and the bra vector is its conjugate transpose. Defining, then, the fundamental

directions for cloud-based quantum-enabled technologies and QOOP.

2. Quantum neural computation and quantum mechanics

j i <sup>0</sup> <sup>¼</sup> <sup>1</sup>

0 

conjugate transpose of 0j i and 1j i, then, we can represent Pauli's operators as:

<sup>σ</sup>^<sup>1</sup> <sup>¼</sup> j i <sup>0</sup> h j <sup>1</sup> <sup>þ</sup> j i<sup>1</sup> h j <sup>0</sup> <sup>¼</sup> 0 1

<sup>σ</sup>^<sup>2</sup> ¼ �ij i <sup>0</sup> h j <sup>1</sup> <sup>þ</sup> <sup>i</sup>j i<sup>1</sup> h j <sup>0</sup> <sup>¼</sup> <sup>0</sup> �<sup>i</sup>

σ^<sup>3</sup> ¼ j i 0 h j� 0 j i1 h j¼ 1

, j i<sup>1</sup> <sup>¼</sup> <sup>0</sup>

with the corresponding bra vectors 0h j and 1h j being defined, respectively, as the

1 

1 0 

> i 0

1 0 0 �1  (1)

(2)

(3)

(4)

classical gamble.

under risk problem, using QUANNs and QOOP.

ket vectors 0j i and 1j i, respectively, as:

96

The unit operator on the two-dimensional Hilbert space, spanned by the basis f g j i <sup>0</sup> ; j i<sup>1</sup> , is denoted by ^<sup>1</sup> <sup>¼</sup> j i <sup>0</sup> h j <sup>0</sup> <sup>þ</sup> j i<sup>1</sup> h j <sup>1</sup> which has the form of the identity matrix.

The Walsh-Hadamard transform unitary operator is, in turn, given by:

$$
\hat{U}\_{WH} = \frac{\hat{\sigma}\_1 + \hat{\sigma}\_3}{\sqrt{2}} = \frac{1}{\sqrt{2}} \begin{pmatrix} 1 & 1 \\ 1 & -1 \end{pmatrix} \tag{5}
$$

We also use the usual notation for the ket vectors j i <sup>þ</sup> <sup>¼</sup> <sup>U</sup>^ WHj i <sup>0</sup> and j i � <sup>¼</sup> <sup>U</sup>^ WHj i<sup>1</sup> .

Besides the above notation, we denote the binary alphabet by A2 ¼ f g 0; 1 and the set of d-length binary strings by A<sup>d</sup> <sup>2</sup> , using boldface letters to represent binary strings of length greater than 1.

Using this notation, we are now ready to address some basic general properties of quantum neural computation.

The basic computational unit of a QUANN is a neuron with a two-level firing dynamics that can be described by the neural firing operator [5, 6]:

$$
\hat{\nu} = \frac{\hat{\mathbf{1}} - \hat{\sigma}\_3}{2} \nu
\tag{6}
$$

where ν is a neural firing frequency expressed in Hertz. The eigenvectors for this operator are given by:

$$
\hat{\nu}|\mathbf{s}\rangle = \mathfrak{s}\nu|\mathfrak{s}\rangle, \mathfrak{s} = \mathbf{0}, \mathbf{1} \tag{7}
$$

Therefore, the eigenvector 0j i corresponds to a neural activity where the firing frequency is 0 Hz, while the eigenvector 1j i corresponds to a neural activity where the firing frequency is 1 Hz. This means that there are two quantized energy levels associated with the artificial neuron, and these energy levels are obtained from the single neuron Hamiltonian, expressed in terms of the neural firing frequency operator as follows [5, 6]:

$$
\hat{H} = \mathfrak{A}\mathfrak{m}\hat{\nu}\tag{8}
$$

Therefore, the eigenvector 0j i is associated with a neural firing energy level of 0 Joules, while the eigenvector 1j i is associated with a neural firing energy level of 2πην Joules.

For a neural network with d neurons, the neural firing activity can be addressed in terms of a neural field in the network, with the firing frequency field operators such that the k-th neuron neural firing operator ν^<sup>k</sup> obeys the eigenvalue equation:

$$
\langle \hat{\nu}\_k | s\_1 s\_2 ... s\_d \rangle = s\_k \nu | s\_1 s\_2 ... s\_d \rangle \tag{9}
$$

and any pair of neural firing operators commute; that is, for k, l ¼ 1, 2, …, d, ν^k; ν^<sup>l</sup> ½ �¼ 0. Thus, the total neural firing frequency operator is given by:

$$
\hat{\boldsymbol{\nu}}\_{\text{Tot}} = \sum\_{k=1}^{d} \hat{\boldsymbol{\nu}}\_{k} \tag{10}
$$

which leads to the eigenvalue spectrum for the neural network:

$$
\hat{\omega}\_{Tot}|s\_1 s\_2...s\_d\rangle = \sum\_{k=1}^d s\_k \nu |s\_1 s\_2...s\_d\rangle \tag{11}
$$

The general computational dynamics of a d-neuron QUANN can be addressed, in the quantum circuit model, by a computational chain of unitary operators, where the networked computation is implemented by conditional unitary operators that follow the structure of the neural links [4–6], which means that not all conditional unitary operators can be implemented in the neural network, but only those that respect the network's topology and processing direction.

Formally, then, an N-length computational chain that propagates forward in a quantum neural computation circuit is comprised of a sequential product of unitary operators:

$$
\hat{\mathbf{C}} = \hat{U}\_N \hat{U}\_{N-1} \dots \hat{U}\_1 \tag{12}
$$

s 0 ρ^out j js <sup>0</sup> D E <sup>s</sup>

s 0 ∈ A<sup>d</sup>

chain:

pattern r 0

r 0 ¼ s 0

99

s 0 ρ^out j js <sup>0</sup> D E <sup>¼</sup> <sup>∑</sup>

0 � � E s <sup>0</sup> D �

� ¼ ∑ <sup>r</sup>, <sup>s</sup><sup>∈</sup> A<sup>d</sup> 2

DOI: http://dx.doi.org/10.5772/intechopen.84149

Quantum Neural Machine Learning: Theory and Experiments

<sup>r</sup>, <sup>s</sup><sup>∈</sup> A<sup>d</sup> 2

returning to Eq. (15), each amplitude r

, and the reverse amplitude <sup>s</sup> <sup>U</sup>^ †

When the two output firing patterns do not match, r

structure of a general density operator for a quantum system [6].

ρ^ ¼ ∑ m, <sup>n</sup>

The off-diagonal components of such an operator are such that there is no matching between the corresponding eigenvalues, only in the diagonal do we find a matching between the eigenvalues. The ket vector can, in this case, be considered as a probing vector, while the bra vector can be considered as a response vector. In this way, only when a probed alternative eigenvalue finds a matching response eigenvalue do we have an echo for an alternative eigenvalue that can be

response that links the output firing pattern s

ρr, <sup>s</sup> s 0 �

ρr, <sup>s</sup> s 0 �

<sup>U</sup>^ NU^ <sup>N</sup>�1…U^ <sup>1</sup>

This means that the neural field computes each alternative final firing pattern

<sup>2</sup> with a projective intensity given by the weighted sum over each pair of alternative initial firing patterns propagated in both directions of the computational

<sup>U</sup>^ NU^ <sup>N</sup>�1…U^ <sup>1</sup>

0

� � �

response that comes from the end of the computational circuit to the beginning, a

between the probe and the response, and when the two firing patterns match,

, an echo is produced with an intensity given by the sum in Eq. (17); the computation is, then, like the search for the solution to a computational problem, where each probed alternative final output gets a response with a specific intensity. These dynamics are simultaneous, that is, the QUANN processes in both the forward and backward directions simultaneously to arrive at the final result.

The above fundamental computational dynamics is characteristic of quantum mechanics, and not limited to QUANNs or quantum computation, nor is it dependent on one's interpretation of quantum mechanics. It arises when one considers the

Indeed, as an example, let us consider a general density operator for a quantized observable O^ on some quantum system, which, for the purpose of illustration we consider to have a discrete, not necessarily finite, non-degenerate eigenvalue spectrum, so that the ket vectors j i <sup>m</sup> , for <sup>m</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>…, satisfying O m^ j i <sup>¼</sup> omj i <sup>m</sup> , span the basis for a Hilbert space associated with the quantum system with respect to the observable; then, the general dynamics for the quantum system, with respect to the observable, can be represented as a density operator on the system's Hilbert space:

�

probing computational dynamics from the beginning to the end of the computational circuit that links the initial (input) firing pattern r to the final probed (output) firing

<sup>1</sup>…U^ †

0

� �

D E <sup>s</sup> <sup>U</sup>^ †

From a computer science standpoint, this two-directional propagation, which is a basic result of the quantum circuit-based computation (generalizable to any type of quantum computer), exhibits a form of forward propagation and backward propagation, where the forward and backward amplitudes can be, from a computer science standpoint, addressed in terms of a probe and response dynamics, respectively;

� �

D E <sup>s</sup> <sup>U</sup>^ †

�r

<sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup>

� �

<sup>N</sup>�<sup>1</sup>U^ † N

� � � s <sup>0</sup> D E can be addressed as a

> 0 6¼ s 0

�r

<sup>0</sup> D E ! <sup>s</sup>

<sup>1</sup>…U^ †

� � �

<sup>1</sup>…U^ †

�<sup>r</sup> � � can be addressed as a

� � �

<sup>N</sup>�1U^ † N

to the initial input firing pattern s.

ρm,nj i m h j n (18)

, we have a mismatch

� � � s <sup>0</sup> D E (17)

<sup>N</sup>�1U^ † N � � � s 0 � � E s <sup>0</sup> D � �

(16)

The sequence is read from right to left and such that U^ <sup>1</sup> is the first operator to be applied and U^ <sup>N</sup> is the last. This is the forward sequence proceeding from the beginning to the end of the computation. The reverse chain, which propagates backward in the computational circuit, is, then, given by the conjugate transpose of the forward chain:

$$
\hat{\mathbf{C}}^{\dagger} = \hat{\mathbf{U}}\_1^{\dagger} \dots \hat{\mathbf{U}}\_{N-1}^{\dagger} \hat{\mathbf{U}}\_N^{\dagger} \tag{13}
$$

Formally, given a general initial density operator, representing the initial neural field dynamics of the QUANN, expressed as follows:

$$\hat{\rho}\_{in} = \sum\_{\mathbf{r}\_{\mathbf{r}}, \mathbf{s} \in A\_2^d} \rho\_{\mathbf{r}, \mathbf{s}} |\mathbf{r}\rangle\langle\mathbf{s}| \tag{14}$$

the quantum computation can be addressed in terms of the propagation:

$$\begin{split} \hat{\rho}\_{\text{out}} &= \hat{\mathsf{C}} \hat{\rho} \hat{\mathsf{C}}^{\dagger} = \\ &= \sum\_{\mathbf{r}', \mathbf{s}' \in \mathsf{A}\_{2}^{d}} \left( \sum\_{\mathbf{r}, \mathbf{s} \in \mathsf{A}\_{2}^{d}} \rho\_{\mathbf{r}, \mathbf{s}} \Big< \Big/ \hat{\boldsymbol{\mathcal{U}}}\_{N} \hat{\boldsymbol{\mathcal{U}}}\_{N-1} ... \hat{\boldsymbol{\mathcal{U}}}\_{1} \Big| \mathbf{r} \Big> \Big/ \mathbf{s}' \Big| \Big< \Big/ \mathbf{s} \Big| \Big/ \mathbf{s} \Big| \hat{\boldsymbol{\mathcal{U}}}\_{1}^{\dagger} ... \hat{\boldsymbol{\mathcal{U}}}\_{N-1}^{\dagger} \hat{\boldsymbol{\mathcal{U}}}\_{N}^{\dagger} \Big| \mathbf{s}' \Big> \Big) \right. \end{split} \tag{15}$$

The firing patterns, in Eq. (15), r and s correspond to input neural firing patterns for the QUANN, while the firing patterns r 0 and s 0 correspond to output neural firing patterns; in this way, the quantum neural computation is propagating in both directions of the computational chain, so that we have the propagation from the input to the output (from the beginning to the end of the computational chain), which corresponds to the amplitude r 0 <sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup> � � � �<sup>r</sup> � �, and the propagation from the output to the input (from the end to the beginning of the computational chain), which corresponds to the amplitude <sup>s</sup> <sup>U</sup>^ † <sup>1</sup>…U^ † <sup>N</sup>�<sup>1</sup>U^ † N � � � � � � s <sup>0</sup> D E.

For the cases where there is a mismatch between the final output firing dynamics, that is, when r 0 6¼ s 0 , the QUANN does not reach a well-defined output; from a computational perspective, we can state that the network does not reach a final solution, since the output computed in the forward direction of the computational circuit, r 0 <sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup> � � � �<sup>r</sup> � �, does not match the output computed in the reverse direction of the computational circuit, <sup>s</sup> <sup>U</sup>^ † <sup>1</sup>…U^ † <sup>N</sup>�<sup>1</sup>U^ † N � � � � � � s <sup>0</sup> D E.

However, when r 0 ¼ s 0 , the output computed in the forward and backward directions matches; this leads to the diagonal components of the final density operator that, for each s 0 ∈ A<sup>d</sup> <sup>2</sup> , are given by:

Quantum Neural Machine Learning: Theory and Experiments DOI: http://dx.doi.org/10.5772/intechopen.84149

The general computational dynamics of a d-neuron QUANN can be addressed, in the quantum circuit model, by a computational chain of unitary operators, where the networked computation is implemented by conditional unitary operators that follow the structure of the neural links [4–6], which means that not all conditional unitary operators can be implemented in the neural network, but only those that

Formally, then, an N-length computational chain that propagates forward in a quantum neural computation circuit is comprised of a sequential product of unitary

The sequence is read from right to left and such that U^ <sup>1</sup> is the first operator to be

<sup>1</sup>…U^ †

Formally, given a general initial density operator, representing the initial neural

�r

The firing patterns, in Eq. (15), r and s correspond to input neural firing patterns

0 and s 0

<sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup>

<sup>1</sup>…U^ †

For the cases where there is a mismatch between the final output firing dynam-

�<sup>r</sup> � �, does not match the output computed in the reverse

� � �

<sup>1</sup>…U^ †

<sup>0</sup> D E

� �

<sup>0</sup> D E

<sup>N</sup>�<sup>1</sup>U^ † N � � � s

, the QUANN does not reach a well-defined output; from a

<sup>N</sup>�<sup>1</sup>U^ † N

, the output computed in the forward and backward

� � � s

.

firing patterns; in this way, the quantum neural computation is propagating in both directions of the computational chain, so that we have the propagation from the input to the output (from the beginning to the end of the computational chain),

the output to the input (from the end to the beginning of the computational chain),

0

�

� � �

computational perspective, we can state that the network does not reach a final solution, since the output computed in the forward direction of the computational

directions matches; this leads to the diagonal components of the final density

<sup>2</sup> , are given by:

<sup>0</sup> D E !

r 0 � � E s <sup>0</sup> D � � <sup>s</sup> <sup>U</sup>^ †

<sup>N</sup>�<sup>1</sup>U^ †

applied and U^ <sup>N</sup> is the last. This is the forward sequence proceeding from the beginning to the end of the computation. The reverse chain, which propagates backward in the computational circuit, is, then, given by the conjugate transpose of

<sup>C</sup>^ † <sup>¼</sup> <sup>U</sup>^ †

ρ^in ¼ ∑ r, s∈ <sup>A</sup><sup>d</sup> 2

<sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup>

D E

� �

the quantum computation can be addressed in terms of the propagation:

<sup>C</sup>^ <sup>¼</sup> <sup>U</sup>^ NU^ <sup>N</sup>�1…U^ <sup>1</sup> (12)

<sup>N</sup> (13)

ρr, <sup>s</sup>j ir h j s (14)

<sup>1</sup>…U^ †

�<sup>r</sup> � �, and the propagation from

.

� � �

<sup>N</sup>�<sup>1</sup>U^ † N

correspond to output neural

� � � s

(15)

respect the network's topology and processing direction.

Artificial Intelligence - Applications in Medicine and Biology

field dynamics of the QUANN, expressed as follows:

ρr, <sup>s</sup> r 0 �

operators:

the forward chain:

<sup>ρ</sup>^out <sup>¼</sup> <sup>C</sup>^ρ^C^ † <sup>¼</sup>

¼ ∑ r0 , s<sup>0</sup> ∈ <sup>A</sup><sup>d</sup> 2

ics, that is, when r

0

�

However, when r

operator that, for each s

circuit, r

98

∑ r, s∈ <sup>A</sup><sup>d</sup> 2

for the QUANN, while the firing patterns r

which corresponds to the amplitude r

0 6¼ s 0

<sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup>

� �

which corresponds to the amplitude <sup>s</sup> <sup>U</sup>^ †

direction of the computational circuit, <sup>s</sup> <sup>U</sup>^ †

0 ∈ A<sup>d</sup>

0 ¼ s 0

$$\left\langle \left\langle \mathbf{s'} \middle| \hat{\rho}\_{\text{out}} \middle| \mathbf{s'} \right\rangle \middle| \mathbf{s'} \right\rangle \middle| \mathbf{s'} \right\rangle = \left( \sum\_{\mathbf{r}, \mathbf{s} \in A\_2^d} \rho\_{\mathbf{r}, \mathbf{s}} \left\langle \mathbf{s'} \middle| \hat{U}\_N \hat{U}\_{N-1} \dots \hat{U}\_1 \middle| \mathbf{r} \right\rangle \middle\langle \mathbf{s} \middle| \hat{U}\_1^\dagger \dots \hat{U}\_{N-1}^\dagger \hat{U}\_N^\dagger \middle| \mathbf{s'} \right\rangle \right) \left| \mathbf{s'} \right\rangle \left\langle \mathbf{s'} \middle| \right| \tag{16}$$

This means that the neural field computes each alternative final firing pattern s 0 ∈ A<sup>d</sup> <sup>2</sup> with a projective intensity given by the weighted sum over each pair of alternative initial firing patterns propagated in both directions of the computational chain:

$$\left\langle \mathbf{s}' \middle| \hat{\rho}\_{\text{out}} \middle| \mathbf{s}' \right\rangle = \sum\_{\mathbf{r}\_{2} \mathbf{s} \in \mathcal{A}\_{2}^{d}} \rho\_{\mathbf{r}, \mathbf{s}} \left\langle \mathbf{s}' \middle| \hat{U}\_{N} \hat{U}\_{N-1} \dots \hat{U}\_{1} \middle| \mathbf{r} \right\rangle \left\langle \mathbf{s} \middle| \hat{U}\_{1}^{\dagger} \dots \hat{U}\_{N-1}^{\dagger} \hat{U}\_{N}^{\dagger} \middle| \mathbf{s}' \right\rangle \tag{17}$$

From a computer science standpoint, this two-directional propagation, which is a basic result of the quantum circuit-based computation (generalizable to any type of quantum computer), exhibits a form of forward propagation and backward propagation, where the forward and backward amplitudes can be, from a computer science standpoint, addressed in terms of a probe and response dynamics, respectively; returning to Eq. (15), each amplitude r 0 <sup>U</sup>^ NU^ <sup>N</sup>�<sup>1</sup>…U^ <sup>1</sup> � � � �<sup>r</sup> � � can be addressed as a probing computational dynamics from the beginning to the end of the computational circuit that links the initial (input) firing pattern r to the final probed (output) firing pattern r 0 , and the reverse amplitude <sup>s</sup> <sup>U</sup>^ † <sup>1</sup>…U^ † <sup>N</sup>�<sup>1</sup>U^ † N � � � � � � s <sup>0</sup> D E can be addressed as a response that comes from the end of the computational circuit to the beginning, a response that links the output firing pattern s 0 to the initial input firing pattern s.

When the two output firing patterns do not match, r 0 6¼ s 0 , we have a mismatch between the probe and the response, and when the two firing patterns match, r 0 ¼ s 0 , an echo is produced with an intensity given by the sum in Eq. (17); the computation is, then, like the search for the solution to a computational problem, where each probed alternative final output gets a response with a specific intensity.

These dynamics are simultaneous, that is, the QUANN processes in both the forward and backward directions simultaneously to arrive at the final result.

The above fundamental computational dynamics is characteristic of quantum mechanics, and not limited to QUANNs or quantum computation, nor is it dependent on one's interpretation of quantum mechanics. It arises when one considers the structure of a general density operator for a quantum system [6].

Indeed, as an example, let us consider a general density operator for a quantized observable O^ on some quantum system, which, for the purpose of illustration we consider to have a discrete, not necessarily finite, non-degenerate eigenvalue spectrum, so that the ket vectors j i <sup>m</sup> , for <sup>m</sup> <sup>¼</sup> <sup>0</sup>, <sup>1</sup>, <sup>2</sup>…, satisfying O m^ j i <sup>¼</sup> omj i <sup>m</sup> , span the basis for a Hilbert space associated with the quantum system with respect to the observable; then, the general dynamics for the quantum system, with respect to the observable, can be represented as a density operator on the system's Hilbert space:

$$
\hat{\rho} = \sum\_{m,n} \rho\_{m,n} |m\rangle\langle n| \tag{18}
$$

The off-diagonal components of such an operator are such that there is no matching between the corresponding eigenvalues, only in the diagonal do we find a matching between the eigenvalues. The ket vector can, in this case, be considered as a probing vector, while the bra vector can be considered as a response vector.

In this way, only when a probed alternative eigenvalue finds a matching response eigenvalue do we have an echo for an alternative eigenvalue that can be actualized, and the probability for this actualization coincides with the diagonal density value ρm,m which corresponds to the echo intensity. This is a basic result from quantum mechanics that extends to any observable, including observables with both discrete as well as continuous spectra.

It is important to stress that this echo dynamics is not specific to QUANNs, but is present in any quantum system; any density operator characterizing a quantum system exhibits, in the formalism, this main dynamics, so the echo dynamics is a characteristic of the physics of quantum systems and accounts for Born's probability rule in quantum mechanics—that is, the probability of an alternative eigenvalue to be observed is equal to the corresponding diagonal component of a density operator.

Therefore, embedded within quantum mechanics' formalism, we find an account of Born's probability rule. Furthermore, given a Hamiltonian operator for the quantum system H^ <sup>S</sup>, and a time lapse of Δt, quantum mechanics defines the unitary propagation of a density operator at time t<sup>0</sup> as:

$$
\hat{\rho}(t\_0 + \Delta t) = e^{-\frac{i}{\hbar}\hat{H}\_S\Delta t}\hat{\rho}(t\_0)e^{\frac{i}{\hbar}\hat{H}\_S\Delta t} \tag{19}
$$

methods, and how the mathematics is built and applied for prediction of experimental results. The interpretations do not stem from any ambiguity or lack of robustness in the formalism and in the application of the formalism, they stem from the fact that not everything is accounted for by the formalism, and that is where the

dealing with, what it explains, and what is outside its theoretical scope.

To better frame this issue, one must consider the nature of the theory that one is

Quantum mechanics is, in fact, a probabilistic theory of the quantized dynamics of fundamental physical fields, fields that work at the level of the building blocks of physical nature. The physical theory and methods that form the basic structure of quantum mechanics developed progressively from empirical observations and statistical findings on fundamentally random outcomes of physical experiments deal-

This means that physicists found the basic rules for (dynamical) probability assignments that robustly capture the main probabilistic dynamics of quantum fields. To understand the nature of the theory, it is important to stress that it was born out of laboratory experiments, that it was built out of the statistical patterns found in an observed stochastic dynamics, and that it was aimed at predicting the statistical distributions of that stochastic dynamics. The current formulation of quantum mechanics essentially encompasses a set of rules for obtaining the probabilities

The theory does not state anything beyond that. A point that allowed many physicists to pragmatically take the theory as it is, not dwelling on the why quantum systems work that way, that is, to take the theory as a rule book that works, is robustly tested empirically, applying it to problems following what is usually called

When one starts to ask on the why quantum systems work that way, the interpretations enter into play, but they go beyond the physical setting of the theory in the sense that they are related to ontological questions; that is, each interpretation regards the ontological issue of physical reality and why the quantum dynamics

In the pragmatic stance, one just takes the formalism as a recipe, calculates the echo intensities without dwelling further on it. Any result in quantum mechanics applying the formalism is valid and empirically testable and the formalism has time and again, during twentieth and twenty-first centuries, been shown to be robust in

One way out of the ontological questioning would be to assume that we are dealing with human representations, that we cannot speak of a reality independent of human representations and experiments, that is, that the question of what reality really is outside those representations and experiments cannot be answered and, therefore, one just postulates that the field follows the echoes. This was the approach of the Copenhagen school, including Bohr and Born, leading to Born's rule that the probabilities are coincident with the echoes, a rule that is introduced, usually, in quantum mechanics' classes as a postulate, a very detailed description of this can be

Contrasting with the Copenhagen school are the ontological schools, so called

Quantum mechanics itself does not state anything about this, so there is room for proposals; Cramer [7], for instance, considers these interpretations as actually new physical theories that go beyond the strict formalism and introduce new conjecture that cannot be tested under the formalism itself. The ontological interpretations that include the Bohmian and Everettian lines are all consistent with the

because they assume a reality independent of human representations and

follows the echoes with probabilities coincident with the echo intensities.

interpretations come in.

ing with the quantum level.

a shut up and calculate stance.

its predictions.

found in [7].

experiments.

101

associated with the dynamics of quantum systems.

Quantum Neural Machine Learning: Theory and Experiments

DOI: http://dx.doi.org/10.5772/intechopen.84149

In the case of the illustrative general example, given in Eq. (18), we get:

$$
\hat{\rho}\left(t\_0 + \Delta t\right) = \sum\_{m\_2 \neq 0} \left( \sum\_{k,l} \rho\_{k,l}(t\_0) \langle m | e^{-\frac{i\hat{H}\_S \Delta t}{\hbar}} |k\rangle \langle l | e^{\frac{i}{\hbar} \hat{H}\_S \Delta t} |n\rangle \right) |m\rangle\langle n| \tag{20}
$$

where <sup>m</sup> exp �i=ηH^ <sup>S</sup>Δ<sup>t</sup> � � � � � �<sup>k</sup> � � is a forward in time propagating amplitude from the k-th initial eigenvalue to the m-th final eigenvalue and <sup>l</sup> exp <sup>i</sup>=ηH^ <sup>S</sup>Δ<sup>t</sup> � � � � � �<sup>n</sup> � � is a backward in time propagating amplitude from the n-th final eigenvalue to the l-th initial eigenvalue<sup>1</sup> , and this basic dynamics is a general result that stems from Schrödinger's unitary evolution.

Cramer was, however, the first to fully address the consequences of this dynamics and propose the concept of echo, within the context of quantum mechanics, addressing it related to Born's rule, deriving Born's rule from within the quantum formalism.

While Cramer [7] addresses the echo in terms of the encounter of a forwardpropagating retarded wave (which we addressed above under the probe dynamics, proceeding forward from the beginning to the endpoint of the unitary evolution) and the backward-propagating advanced wave (which we addressed above under the response dynamics, proceeding from the endpoint to the beginning of the unitary evolution), by working with the density operator, instead of the wave function, we get a clearer picture of the corresponding dynamics, which accounts, in the case of any quantum physical system, for both the off-diagonal terms (as failed echoes) and the diagonal terms of the density operator (as the echoes where the probe was met by a matching response) with the echo intensity giving Born's probability rule. This result is generalizable and independent of the interpretation of quantum mechanics that one follows; that is, all interpretations of quantum mechanics agree with the above results.

It is important to clarify what an interpretation of quantum mechanics is and why there are different interpretations of the same theoretical body and equations. It turns out that the main interpretations do not disagree on the formalism,

<sup>1</sup> One may notice the change in the time lapse signal so that the conjugate transposition corresponds to time reversal.
