#### 2.1. “Quantum” Measurement of a Given String within a Superposition: “Collapse of the System Wavefunction”

Any of the

M-long strings

$X(1),X(2)\dots X(M)$ in an

M-noise-bit INBL system is using half of the reference wires to form its signal:

We can reduce an arbitrary superposition to any of its product-string elements by “grounding” (forcing a zero signal amplitude onto) the inverse (that is, other half) of the reference wires, namely, the reference wires of the bit values

$\overline{X}(1),\overline{X}(2)\dots \overline{X}(M)$, while keeping the rest of the reference wires, the

$X(1),X(2)\dots X(M)$ wires, at their reference sources. For the required hardware, see

Figure 7, and for a practical example, see

Figure 8. Any other string in the arbitrary superposition uses at least one of the grounded bit values, thus the signal components all these product-strings become zero in the superposition. This situation corresponds to the “collapse of the system wavefunction” in a quantum system. If the superposition signal amplitude is not zero during the clock period when the measurement is carried out, and after the grounding the signal becomes zero, that is a proof that the product-string was not in the superposition. Otherwise, the new signal will be a non-zero signal

${R}_{1X(1)}(t){R}_{2X(2)}(t)\dots {R}_{MX(M)}(t)$ of the product-string.

Therefore, grounding the inverse reference wires of a string and observing the remaining signal of the superposition represents the quantum-type measurement in instantaneous noise-based logic. Note, there are other types of measurements, too, but they are of statistical nature, while grounding the reference wires yields a deterministic measuremen, provided the instantaneous amplitude of the original superposition is non-zero at that time, that is, during the clock-period of the measurement.

It is important to note that, even though this measurement looks similar to quantum measurements, it is

very different from quantum measurements.

Quantum measurements are statistical, which means that they

contain errors, and furthermore they also

require an exponential complexity when the task is to measure a string in an exponentially large superposition. For example, in a photonic quantum system with

M qubits, the Universe consists of the superposition of 2

^{M} different strings. If in this uniform (flat) superposition, we want to detect a single string with acceptable error probability, the number

P of available photons in the system must satisfy:

implying an

exponential time complexity in the measurement system of filters/beamsplitters and coincidence detectors with finite time-resolution. On the other hand, in INBL, the same measurement takes just a single time step.

#### 2.2. Deterministic Measurement and Evaluation of Entangled States and Their Superposition in INBL

Suppose Alice sets up a two-noise-bit system (

M = 2) with a superposition signal

S(

t) containing entanglement (similar to the Einstein–Rosen–Podolsky [

1] problem’s superposition and the related Bell states):

This 0(1),1(2) + 1(1),0(2) superposition is unknown by Bob except in the conditions that it consists of the sum of any of the possible ${R}_{ij}(t){R}_{pq}(t)$ two-bit strings $i,p\in \left\{1,2\right\};\text{}j,q\in \left\{0,1\right\}$, with the restriction that a given value of a given bit can exist at most in a single string. For example, alternative superpositions could be 0(1),0(2) + 1(1),1(2), that is, ${R}_{10}(t){R}_{20}(t)+{R}_{11}(t){R}_{21}(t)$; or single strings such as 0(1),1(2), that is, ${R}_{10}(t){R}_{21}(t)$; 1(1),0(2), that is, ${R}_{11}(t){R}_{20}(t)$; 0(1),0(2), that is, ${R}_{10}(t){R}_{20}(t)$; or 1(1),1(2), that is, ${R}_{11}(t){R}_{21}(t)$.

Bob’s job is to measure

S(

t) and decide which is the actual superposition from these different possibilities. To identify strings, statistical methods exist, e.g., [

9,

11,

12,

14] and some of them, like the time-shifted RTW scheme [

9] and the Stacho [

11] method, require only polynomial computation complexity versus the number of noise-bits. However, these methods used on superposition would require

exponential complexity with expanding

M (except when the job is to identify a single string and the rest of the superposition is known [

14]).

Here we show that, by utilizing operations on the reference wires, the measurement and analysis can be done in a fast way, which is deterministic, with polynomial complexity. The procedure takes place during a clock time t when the superposition signal amplitude is non-zero, that is, $S(t)\ne 0$:

(i) Bob first checks the existence of the reference signal

${R}_{10}(t)$ in the superposition

$S(t)\ne 0$ by grounding (forcing zero on) the reference wire of the

inverse bit value of

${R}_{10}(t)$, that is, forcing

${R}_{11}(t)=0$, and then checking if the resulting superposition signal

${S}_{2}(t)$ is zero. In the above example, see Equation (7), the superposition gets reduced to:

which means

${S}_{2}(t)\ne 0$.

In alternative cases, when the grounding causes

${S}_{2}(t)=0$, Bob jumps to step (iii) below. In the present situation,

${S}_{2}(t)\ne 0$ (see Equations (7) and (8)), Bob needs to evaluate which value of the second bit is entangled with the 0 value of the first bit. This string can be evaluated in various known statistical ways, e.g., [

9,

11,

14] but here we again propose a deterministic tool by utilizing the reference wires:

(ii) While still being in the same clock period t and keeping ${R}_{11}(t)=0$, Bob also grounds the 0 value of the second bit, ${R}_{20}(t)=0$, and gets the superposition ${S}_{3}(t)$. If ${S}_{3}(t)=0$, then ${R}_{20}(t)$ is the entangled bit signal. If ${S}_{3}(t)\ne 0$, then ${R}_{21}(t)$ is the bit value in question, which is the case in the present example.

Note, Bob can proceed in an alternative way: He grounds the reference ${R}_{21}(t)$, ${R}_{21}(t)=0$, while keeping the grounding ${R}_{11}(t)=0$. If ${S}_{3}(t)=0$, then ${R}_{21}(t)$ represents the entangled bit value.

(iii) Now, Bob repeats the above procedures (i–ii) with swapped bit values. He first checks the existence of the reference signal ${R}_{11}(t)$ in the superposition by grounding the reference wire of the inverse bit value (forcing ${R}_{10}(t)=0$). If the reference signal ${R}_{11}(t)$ is not in the original superposition (Equation (7)), then ${S}_{2}(t)=0$ and Bob has completed the evaluation. Otherwise, he proceeds with the swapped bit values as described in the above procedure (i–ii).