Quantum autonomous Boolean networks

TL;DR

This paper introduces a quantum version of autonomous Boolean networks that retains classical properties while also displaying unique quantum behaviors, expanding the modeling capabilities for complex systems.

Contribution

The paper develops a quantum autonomous Boolean network model that combines classical dynamics with quantum properties, offering new insights into complex system modeling.

Findings

Exhibits classical-like spontaneous order

Demonstrates uniquely quantum behaviors

Shows a rich landscape of dynamical behaviors

Abstract

Boolean networks, first developed in the late 1960s as a tool for studying complex disordered dynamical systems, consist of nodes governed by Boolean functions whose evolution is entirely deterministic in that the state of the network at a given time fully determines the state of the network at some future time. They are known for exhibiting a high degree of spontaneous order and have since become a fundamental tool for modeling a wide variety of systems. In this article I develop a model for quantum autonomous Boolean networks that exhibits many of the same properties as the classical model while also demonstrating uniquely quantum properties within a rich landscape of behavior.

Tables

Table 1. Table 1: As the truth table for the XOR function indicates, each pair of outputs can uniquely be identified with a pair of inputs which means the XOR function is reversible and thus does not require an ancilla. There is ambiguity in the OR function since both ( x , y ) = ( 1 , 0 ) 𝑥 𝑦 1 0 (x,y)=(1,0) and ( x , y ) = ( 1 , 1 ) 𝑥 𝑦 1 1 (x,y)=(1,1) as inputs lead to ( x , y ⊕ f ​ ( x ) ) = ( 1 , 1 ) 𝑥 direct-sum 𝑦 𝑓 𝑥 1 1 (x,y\oplus f(x))=(1,1) as an output. As such the OR function is not reversible and thus requires an ancilla in order to be represented reversibly.

XOR					OR
$x$	$y$	$x$	$y\oplus f\left(x\right)$		$x$	$y$	$x$	$y\oplus f\left(x\right)$
0.5pt]1-4 0.5pt]6-9 0	0	0	0		0	0	0	0
0	1	0	1		0	1	0	1
1	0	1	1		1	0	1	1
1	1	1	0		1	1	1	1

Table 2. Table 2: In order to reversibly represent the OR function, y 𝑦 y must be an ancilla to which the logical output is mapped.

OR
$x_1$	$x_2$	$y$	$x_1$	$x_2$	$y\oplus f\left(x\right)$
[0.5pt] 0	0	0	0	0	0
0	1	0	0	1	1
1	0	0	1	0	1
1	1	0	1	1	1

Table 3. Table 3: If we attempt to simultaneously implement both the AND and the OR function on the same set of three inputs, we find that the process is not reversible since every unique output string could have arisen from one of two possible input strings.

				AND	OR
$x_1$	$x_2$	$y$	$x_1$	$x_2\oplus f_1(x_1,y)$	$y\oplus f_2(x_1,x_2)$
[0.5pt] 0	0	0	0	0	0
1	0	0	1	0	1
0	1	0	0	0	1
0	0	1	0	0	0
1	0	1	1	1	1
1	1	0	1	0	1
0	1	1	0	0	1
1	1	1	1	1	1

Table 4. Table 4: The truth table for a classical Boolean network of three variables whose values are governed by the AND, OR, and OR functions respectively is shown here.

			AND	OR	OR
			$x_1\oplus$	$x_2\oplus$	$x_3\oplus$
$x_1$	$x_2$	$x_3$	$f_1(x_2,x_3)$	$f_2(x_1,x_3)$	$f_3(x_1,x_2)$
[0.5pt] 0	0	0	0	0	0
1	0	0	0	1	1
0	1	0	0	0	1
0	0	1	0	1	0
1	0	1	0	1	1
1	1	0	0	1	1
0	1	1	1	1	1
1	1	1	1	1	1

Table 5. Table 5: The evolution of the first seven variables in a hypothetical seven-variable network shows a frozen core corresponding to the fourth element which gives two functionally isolated islands consisting of ( x 1 , x 2 , x 3 subscript 𝑥 1 subscript 𝑥 2 subscript 𝑥 3 x_{1},x_{2},x_{3} ) and ( x 5 , x 6 , x 7 subscript 𝑥 5 subscript 𝑥 6 subscript 𝑥 7 x_{5},x_{6},x_{7} ) respectively.

$t$	$x_1$	$x_2$	$x_3$	$x_4$	$x_5$	$x_6$	$x_7$
[0.5pt] 1	0	1	0	1	0	0	1
2	1	0	1	1	1	1	0
3	0	1	1	1	1	0	1
4	1	1	1	1	0	1	0
5	0	1	0	1	1	1	1
6	1	0	0	1	1	0	0
7	0	0	1	1	0	1	1
8	1	0	1	1	0	1	0
9	1	1	0	1	1	1	1
10	0	1	0	1	1	0	1

Equations

$$|\mathbf{x}⟩\mapsto (-1{)}^{f(\mathbf{x})}|\mathbf{x}⟩$$

$$|\mathbf{x}⟩|\mathbf{y}⟩\mapsto U_f|\mathbf{x}⟩|\mathbf{y}⟩\equiv |\mathbf{x}⟩|\mathbf{y}\oplus f(\mathbf{x})⟩$$

$$(x_1,x_2,y)\mapsto (x_1,x_2,y\oplus f(x_1,x_2)).$$

$$(x_1,x_2,y)\mapsto (x_1,x_2\oplus f_1(x_1,y),y\oplus f_2(x_1,x_2))$$

$$(x_1,x_2,y_1,y_2)\mapsto (x_1,x_2,y_1\oplus f_1(x_1,x_2),y_2\oplus f_2(x_1,x_2))$$

$$|x_1,x_2⟩|y_1,y_2⟩\mapsto U_{f_1}{U}_{f_2}|x_1,x_2⟩|y_1,y_2⟩.$$

$$|x_1,x_2⟩|y_1,y_2⟩\mapsto U_{f_2}{U}_{f_1}|x_1,x_2⟩|y_1,y_2⟩$$

$$\rho (\mathbf{x},\mathbf{y})\equiv |\mathbf{x}⟩|\mathbf{y}⟩⟨\mathbf{y}|⟨\mathbf{x}|$$

$$\rho (\mathbf{x},\mathbf{y}\oplus f(\mathbf{x}))=U_f\rho (\mathbf{x},\mathbf{y})U_f^{†}.$$

$$\rho (\mathbf{X},\mathbf{Y})$$

$$U_F$$

$$F(\mathbf{X})$$

$$\rho (\mathbf{X},\mathbf{Y}\oplus F(\mathbf{X}))=U_F\left[\rho (\mathbf{X},\mathbf{Y})\right]U_F^{†}.$$

$$\begin{array}[]{rlcrl}\rho(0)&=\left|0\right>\left<0\right|&&\rho(1)&=\left|1\right>\left<1\right|\\ \rho(+)&=\left|+\right>\left<+\right|&&\rho(-)&=\left|-\right>\left<-\right|.\end{array}$$

$$I_m(A_1:\cdots :A_n)$$

$$=S(\rho ||{\rho }^{A_1}\otimes \cdots \otimes {\rho }^{A_n})$$

$$\ge 0$$

Full text

I Introduction

Boolean networks were first developed by Kauffman in the late 1960s as a tool for studying complex disordered dynamical systems [1]. His original intent was to find networks that might possess enough order such that they allow for adaptation and selection as in, for example, genetic regulatory processes and similar systems [2]. The nodes of these networks consist of Boolean functions and their evolution is entirely deterministic; the state of the network at a given time $t$ fully determines the state of the network at time $t+1$. As such, these networks are referred to as autonomous. If the initial connections between the nodes are randomly determined, then these networks are also referred to as random Boolean networks. Because these networks have a finite number of nodes and are entirely deterministic, they eventually cycle through a finite number of states, i.e. they exhibit state cycles of finite length where the length depends on the number of functions in the network. In some of these networks, the state of one or more of the nodes are “frozen” in a given state regardless of the length of the state cycle. Such elements are referred to as frozen cores. The frozen cores create “islands” of isolated nodes separated by “percolating walls” such that perturbations to variables in one island have no effect on variables in other islands [3]. Since Kauffman’s introduction, the properties of these networks have been extensively studied [4, 5, 6, 7, 8, 9, 10, 11, 12, 13]. They are considered one of the fundamental modeling tools in the study of complex biological systems [14] and are also considered an important tool in the modeling of stochastic dynamical systems [15, 16, 17]. They serve as the canonical example of mechanisms in integrated information theory and its quantum generalizations [18, 19, 20, 21, 22], and have also found applicability in cryptography [23, 24].

To date, however, little work has been done on quantum extensions or analogs of these networks. By extensions and analogs, I am not referring here to the simulation of these networks on quantum systems, but rather to more direct implementations of these networks as circuits of Boolean functions. This, of course, requires modeling the functions as unitary gates but since not all classical Boolean functions are reversible, their implementation as unitary gates in quantum circuits often requires the use of ancilla qubits [22]. A method for the construction of unitary gates for the implementation of classical Boolean functions was first proposed by Deutsch in 1985 [25]. The more general concept of quantum Boolean functions was developed by Montanaro and Osborne in 2010 [26]. The latter was recently used as a basis for the development of a quantum analog of Kauffman’s original model [27]. This analog, however, left a number of fundamental issues unaddressed. In particular it did not explore the effects of quantization on a number of issues, including the lengths of state cycles and the existence of frozen cores, which are of vital importance to the emergence of order and to the robustness of these networks to random perturbations.

This article thus develops a model framework for such networks that includes these features, allowing for a more direct comparison to Kauffman’s networks and extending the realm of study for such networks to the quantum domain. The framework reveals a rich landscape of behavior unique to quantum systems including highly variable state cycle lengths that do not depend on the number of functions in the network, as well as the existence of correlations between isolated islands across frozen cores. This framework, however, is not intended as a way to model Kauffman’s networks directly using quantum systems. That is, I am not interested here in reproducing the exact sequences of states that are generated by classical autonomous Boolean networks. Rather, I am interested in the behavior of sets of Boolean functions modeled as unitary gates in the quantum domain. So, while it might be possible to recreate the properties of an autonomous Boolean network consisting of a certain set of Boolean functions using a set of quantum logic gates, instead, in the framework presented here, the network’s gates would be direct quantum implementations of the those classical Boolean functions. In other words, there is a one-to-one correspondence in the present work between the number of operations in the network and the number of Boolean functions that are represented.

I begin in Section II with a review of Boolean functions in general before introducing their unitary implementations and a generalized bit oracle for calculating irreversible classical Boolean functions on quantum networks. In Section III I then give an overview of classical autonomous Boolean networks along with a discussion of circuit implementations of such networks before introducing the quantum framework. In Section IV.1 I discuss the important features of these networks, which I refer to as quantum autonomous Boolean networks (qABNs), and compare them to their classical counterparts. Finally, in Section V I discuss some of the unanswered questions and lines of inquiry that might be undertaken to address them.

II Boolean functions

A classical Boolean function is a function of $k$ input variables and $m$ output variables of the form $f:\{0,1\}^{k}\to\{0,1\}^{m}$ where $\{0,1\}$ is the standard Boolean domain and is isomorphic to $\mathbbm{Z}/2\mathbbm{Z}$ where the addition of any two variables $x$ and $y$ is $x\oplus y$. In the quantum domain, each of our $k$ input variables is represented by a qubit. A quantum Boolean function of $k$ qubits is then a unitary operator $f$ on $k$ qubits such that $f^{2}=\mathbbm{1}$.

It is often standard procedure to identify $\{0,1\}$ with $\{+1,-1\}$ by defining $0\equiv+1$ and $1\equiv-1$ [26]. This makes $\{+1,-1\}$, which is also isomorphic to $\mathbbm{Z}/2\mathbbm{Z}$, the multiplicative group of two elements where products are written $xy$. In this paper, unless otherwise specified, I will employ the $\{0,1\}$ convention in order to better demonstrate the consistency with Kauffman’s original ideas.

Consider a single variable $x$ defined on the Boolean domain $\{0,1\}$ whose state is determined by $k$ input variables, each of which is itself defined on the Boolean domain. The number of combinations of states of $k$ inputs is just $2^{k}$. But for each of these $2^{k}$ combinations, a specific Boolean function must, by definition, specify the value of $x$. Since the values of $x$ lie on the Boolean domain, this means that there are a total of $2^{2^{k}}$ Boolean functions of $k$ inputs. For example there are sixteen two-input Boolean functions, including the familiar AND, OR, and NOT.

There are two natural ways of representing classical Boolean functions on quantum systems. These are known as the phase oracle [28]:

[TABLE]

and the bit oracle (also called the standard oracle):

[TABLE]

where $U_{f}^{2}=U_{f}^{{\dagger}}U_{f}=\mathbbm{1}$, $\mathbf{x}\equiv x_{1}x_{2}\cdots x_{k},x_{j}\in\{0,1\}$, and $\mathbf{y}\equiv y_{1}y_{2}\cdots y_{m},y_{i}\in\{0,1\}$. That is, given an input state $\left|\mathbf{x}\right>\left|\mathbf{y}\right>$, $U_{f}$ maps the function’s logical output to $\left|\mathbf{y}\right>$.

II.1 Unitary implementations

The logical output $\left|\mathbf{y}\right>$ may or may not be a set of ancillas depending on the nature of the function. If a function is naturally reversible then no ancilla is needed. But if the function is not reversible then at least one ancilla is required in order to ensure unitarity. To see this, consider the truth tables of the exclusive OR (XOR) and OR functions in Table II.1.