## 1 Introduction: On Noise-Based Logic
### Hyperspace (Hilbert Space) Vectors: Product-Strings  
It is an important feature that the products of these noises form a _hyperspace_ with \(2^{M}\) geometrical dimensions that we describe below. This hyperspace forms the same Hilbert space that full-scale quantum computing is utilizing. Numbers in the NBL system are \(M\)-bit binary numbers represented by product-strings formed by the noises of the corresponding bit values. For example, in the case of \(M=4\), the signal of the binary number 1010 is carried by the noise product \(R_{11}(t)R_{20}(t)R_{31}(t)R_{40}(t)\). Note that the above string corresponds to the vector \(|1010\)\(>\) in the quantum Hilbert space. This 4 noise-bit system can represent \(2^{4}\) different binary numbers similarly to classical logic circuitry of 4 bits bit resolution. However, the major difference is that, similarly to full-scale quantum computers, the INBL system can carry simultaneously all these numbers in a parallel fashion, with a single signal that is the superposition of all these product-strings. INBL operations on these superpositions act simultaneously on each element in this superposition. This situation can lead to significant speedup. Exponentially Large Superpositions with Polynomial Complexity  
Many exponentially large superpositions can be treated by polynomial complexity in INBL, which is also a core feature of full-scale quantum computing. For example, the Universe \(U(t)\), which is the superposition of all the binary numbers, can be created by adding the signals of the low (0) and the high (1) bit values of each noise-bit and then multiplying these sums:  
\[U(t)\ =\ [R_{10}(t)+R_{11}(t)][R_{20}(t)+R_{21}(t)]\ \ldots\ [R_{M0}(t)+R_{M1}(t)]. \tag{2}\]  
Figure 1: Generic noise-based logic hardware scheme. Figure 2: Generic hardware schematics to form a superposition from the reference signals. Even though this quantity can be set up by \(M\) addition and \(M\)-1 multiplication, that is, by \(2M\)-1 elementary algebraic operations, when Equation (2) is expanded, it forms the sum of all the \(2^{M}\) different product-strings (all the binary numbers) this \(M\) noise-bit system can form. For practical solutions for reducing the probability of zero amplitudes of the Universe, see Section 1.3. It is important to note that although the same technique can be applied to deterministic signals, such as with sinusoidal waves as reference signals, such exponentially large superpositions will require exponential time complexity due to the frequency byproducts of multiplication, which results in an exponential growth of the bandwidth. Even though the bandwidth with continuum noises would increase due to the multiplication, linear frequency filters could limit the effect without losing the distinguishability of logic states; thus the polynomial complexity can be kept by using stochastic reference signals. More on this issue is in Section 1.3. Another simple example is the superposition \(Y_{\text{even}}(t)\) of all the even numbers, which can be obtained by skipping the term corresponding to the 1-bit value of the lowest bit:  
\[Y_{\text{even}}(t)\ =\ R_{10}(t)[R_{20}(t)+R_{21}(t)]\ \dots\ [R_{M0}(t)+R_{M1}(t)] \tag{3}\]  
and the superposition \(Y_{\text{odd}}(t)\) of all the odd numbers is obviously just  
\[Y_{\text{odd}}(t)\ =\ U(t)-Y_{\text{even}}(t). \tag{4}\]  
Equation (4) represents an operation with an exponentially large, O(2\({}^{M\text{-}1}\)), input, and an exponentially large, O(2\({}^{M}\)), output that does not overlap with the input. This requires only O(\(M\)) complexity (see in Section 1.3) and it is carried out by a single algebraic operation: Subtraction. Note, full-scale quantum computers provide similar features in their Hilbert space with their unitary operators. The major difference between the quantum and the NBL schemes stems from the accessibility and robustness of the classical physical information, which not only offers, but also requires different approaches in the algorithms.