# Tuning Logical Phi-Bit State Vectors in an Externally Driven Nonlinear Array of Acoustic Waveguides via Drivers’ Phase

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## Abstract

**:**

## 1. Introduction

^{16}-dimensional space) and the nonseparability of coherent superpositions [4]. The applicability of logical phi-bits to the development of a classical, acoustic-based quantum-inspired information processing platform necessitates that phi-bits and their supporting array of waveguides satisfy the DiVincenzo five criteria for the physical construction of a quantum computer [5].

^{N}[4]. The complete Hilbert space of multi-phi-bit systems can be tiled through their representation (i.e., the choice of the basis) and by tuning the system’s driving conditions, such as the drivers’ frequency [6].

## 2. Materials and Methods

## 3. Results

^{3}) longitudinal waves have a wavelength of around 10 cm, making propagation along rod-like waveguides almost one-dimensional [2]. Additionally, the transducers provide appropriate driving and detecting amplitudes at these frequencies, and the linear longitudinal modes of finite-length waveguides are well-defined [2]. Initially, drivers 1 and 3 are in phase, and their relative phase, $\mathsf{\Delta}\theta $, is increased by increments of 2.5° up to 360°. The displacement field at the detection ends of the waveguides is Fourier transformed. The Fourier spectrum includes primary peaks at the two frequencies ${f}_{1}$ and ${f}_{2}$ and secondary peaks at multiple frequencies, $p{f}_{1}+q{f}_{2}$, corresponding to nonlinear phi-bit modes. The complex amplitude of each phi-bit mode at the ends of the three waveguides is used to calculate the phase differences ${\phi}_{12}$ and ${\phi}_{13}$. In Figure 2, we illustrate these phase differences for two phi-bit modes, namely a phi-bit (a) with nonlinear frequency ${f}_{a}=4{f}_{1}-2{f}_{2}$ ({p = 4, q = −2}) and a phi-bit (b) with frequency ${f}_{b}=4{f}_{1}-{f}_{2}$ ({p = 4, q = −1}). In addition to the response to $\mathsf{\Delta}\theta $ of the phase differences ${\phi}_{12}$ and ${\phi}_{13}$, from phi-bits (a) and (b), we have also measured the phases ${\phi}_{12}\left({f}_{1,2}\right)$ and ${\phi}_{13}\left({f}_{1,2}\right)$ for the primary modes observed at the frequencies ${f}_{1}$ and ${f}_{2}$. We have also calculated the quantities ${\phi}_{12}^{0}\left(a\right)=4{\phi}_{12}\left({f}_{1}\right)-2{\phi}_{12}\left({f}_{2}\right)$ and ${\phi}_{13}^{0}\left(a\right)=5{\phi}_{13}\left({f}_{1}\right)-2{\phi}_{13}\left({f}_{2}\right)$, as well as ${\phi}_{12}^{0}\left(b\right)=4{\phi}_{12}\left({f}_{1}\right)-{\phi}_{12}\left({f}_{2}\right)$ and ${\phi}_{13}^{0}\left(b\right)=5{\phi}_{13}\left({f}_{1}\right)-{\phi}_{13}\left({f}_{2}\right)$. These quantities (${\phi}_{12}^{0}$ and ${\phi}_{13}^{0}$, with the superscript 0) would represent the phase differences of the phi-bits if these were simple linear combinations of ${\phi}_{12}$ and ${\phi}_{13}$ in the primary linear modes. The response of the two phi-bits is composed of two separate sets of features. ${\phi}_{12}$ and ${\phi}_{13}$ of both phi-bits (a) and (b) follow the trend of ${\phi}_{12}^{0}\left(a\right)$, ${\phi}_{12}^{0}\left(b\right)$, ${\phi}_{13}^{0}\left(a\right)$, and ${\phi}_{13}^{0}\left(b\right)$ as functions of $\mathsf{\Delta}\theta $. The variations following the linear combinations of primary mode phases will be subsequently called backgrounds. Sharp phase jumps that happen within a few degrees overlap with the backgrounds. These phase jump amount to less than 180°. Similar behaviors are observed for other phi-bit nonlinear modes.

#### 3.1. Model of Nonlinear Logical Phi-Bit and Effect of Drivers’ Phase on Phi-Bit State Vector

#### 3.1.1. Case I: $n=1$ or 3

#### 3.1.2. Case II: $n=2$

#### 3.2. Example of Initialization Using Nonlinear Phase Jumps

^{2}. By driving the physical system with $\mathsf{\Delta}\theta ~{122}^{\mathrm{o}},$ one can realize the two-phi-bit state labeled (1): $\left(\begin{array}{c}-1\\ 1\end{array}\right)\otimes \left(\begin{array}{c}1\\ 1\end{array}\right)$. Similarly, at $\mathsf{\Delta}\theta ~{157}^{\mathrm{o}}$, ${187}^{\mathrm{o}}$, and ${207}^{\mathrm{o}}$, one can initialize the system into the states: (2): $\left(\begin{array}{c}1\\ 1\end{array}\right)\otimes \left(\begin{array}{c}1\\ 1\end{array}\right)$, (3): $\left(\begin{array}{c}-1\\ 1\end{array}\right)\otimes \left(\begin{array}{c}-1\\ 1\end{array}\right)$ and (4): $\left(\begin{array}{c}1\\ 1\end{array}\right)\otimes \left(\begin{array}{c}-1\\ 1\end{array}\right)$, respectively. The two phi-bit states are constructed as the tensor products of single-phi-bit states and are, therefore, separable (i.e., not classically entangled). However, entanglement is not necessary to correlate the two phi-bits, as would be the case with two qubits, in order to enable parallelism in two-bit operations. Here, the two phi-bits (a) and (b) are correlated via the nonlinearity of the elasticity of the physical system. For phi-bits, classical entanglement would only be needed to initialize two different phi-bit states. This can be achieved with the current data by using a different representation (i.e., basis) of the two phi-bit states. We can see that through a single action on the nonlinear array of coupled acoustic waveguides, that is, tuning the drivers’ phase difference $\mathsf{\Delta}\theta ,$ one can parametrically change the states of phi-bits (a) and (b) and, therefore, simultaneously change the four components of the two phi-bit state vector. This action is actually a controllable operation on two phi-bit states. For instance, changing $\mathsf{\Delta}\theta $ from ${122}^{\mathrm{o}}$ to ${157}^{\mathrm{o}}$ results in the following unitary transformation acting on state (1) and producing state (2) to within a global phase of π:

#### 3.3. Example of Initialization Using Phi-Bit Background Phase

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Picture (

**a**) and exploded view (

**b**) of the array of acoustic waveguides coupled with epoxy resin. In (

**b**), we also show the schematic of the experimental system for generating and detecting logical phi-bits. This includes separate signal generators and amplifiers that are used to drive piezoelectric transducers, driving and detecting the transducers attached to the opposite ends of the waveguides by the pressure of three independent rubber bands. A thin layer of honey is used as an ultrasonic coupling agent between the transducers and the waveguide ends. The detected signals enter an oscilloscope via independent input channels for analysis. The waveguides are suspended by thin threads for isolation.

**Figure 2.**Measured ${\phi}_{12}$ and ${\phi}_{13}$, of phi-bits (

**a**) (left) and (

**b**) (right) as functions of the drivers’ phase difference $\mathsf{\Delta}\theta $. The open circles are the experimental results, with the thin line serving as a guide for the eyes. The solid lines represent the quantities ${\phi}_{12}^{0}\left(a\right)=4{\phi}_{12}\left({f}_{1}\right)-2{\phi}_{12}\left({f}_{2}\right)$, ${\phi}_{13}^{0}\left(a\right)=5{\phi}_{13}\left({f}_{1}\right)-2{\phi}_{13}\left({f}_{2}\right)$, ${\phi}_{12}^{0}\left(b\right)=4{\phi}_{12}\left({f}_{1}\right)-{\phi}_{12}\left({f}_{2}\right)$, and ${\phi}_{13}^{0}\left(b\right)=5{\phi}_{13}\left({f}_{1}\right)-{\phi}_{13}\left({f}_{2}\right)$ calculated from the primary modes.

**Figure 4.**Schematic illustration of the amplitude-frequency response of the forced array of coupled waveguides due to self-interaction. The frequency $\sigma =\frac{1}{\epsilon}\left(2{\omega}_{2}-{\omega}_{1}-{\omega}_{0,n}\right)$ is normalized to $\delta {\mathsf{\Gamma}}_{1}$. For illustrative purposes, we have taken $\frac{3\delta}{8{\omega}_{0,n}}=0.5$, ${\left(\delta {\mathsf{\Gamma}}_{2}\right)}^{2}=0.008$, and ${\mu}^{2}=0.00425$. We consider three values for $\mathsf{\Delta}\theta $ = 0, 90, 150°.

**Figure 5.**Schematic illustration of the phase-frequency response of the forced array of coupled waveguides due to self-interaction. The phase is in degrees. At the fixed frequency $\sigma $~3.1, a change in driver phase $\mathsf{\Delta}\theta $ of 90° (horizontal arrow) may take the nonlinear mode (closed circle) into the single-valued state shown as the open circle; that is, a change in $\eta $ of approximately 130° (vertical arrow).

**Figure 6.**$\mathrm{cos}\mathsf{\Delta}{\phi}_{12}$ and $\mathrm{cos}\mathsf{\Delta}{\phi}_{13}$ for phi-bit (a) (solid line) with a nonlinear frequency of ${f}_{a}=4{f}_{1}-2{f}_{2}$ ({p=4,q=$-$2}) and phi-bit (b) (dashed line) with a frequency of ${f}_{b}=4{f}_{1}-{f}_{2}$ ({p=4,q=$-$1}) as functions of the drivers’ phase difference, $\mathsf{\Delta}\theta $. The vertical lines indicated by (1), (2), (3), and (4) correspond to the four possible two phi-bit states, the tensor product of the single phi-bit states $\left(\begin{array}{c}1\\ 1\end{array}\right)$, and $\left(\begin{array}{c}-1\\ 1\end{array}\right)$.

**Figure 7.**Common background phi-bit phases ${\phi}_{12}\left({f}_{1}={\omega}_{1}/2\pi \right)$ (black line) and ${\phi}_{13}\left({f}_{1}={\omega}_{1}/2\pi \right)$ (grey line).

**Figure 8.**Components of a three-phi-bit representation (see text for details) as functions of the drivers’ phase difference.

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**MDPI and ACS Style**

Deymier, P.A.; Runge, K.; Hasan, M.A.; Lata, T.D.; Levine, J.A.
Tuning Logical Phi-Bit State Vectors in an Externally Driven Nonlinear Array of Acoustic Waveguides via Drivers’ Phase. *Quantum Rep.* **2023**, *5*, 325-344.
https://doi.org/10.3390/quantum5020022

**AMA Style**

Deymier PA, Runge K, Hasan MA, Lata TD, Levine JA.
Tuning Logical Phi-Bit State Vectors in an Externally Driven Nonlinear Array of Acoustic Waveguides via Drivers’ Phase. *Quantum Reports*. 2023; 5(2):325-344.
https://doi.org/10.3390/quantum5020022

**Chicago/Turabian Style**

Deymier, Pierre A., Keith Runge, M. Arif Hasan, Trevor D. Lata, and Josh A. Levine.
2023. "Tuning Logical Phi-Bit State Vectors in an Externally Driven Nonlinear Array of Acoustic Waveguides via Drivers’ Phase" *Quantum Reports* 5, no. 2: 325-344.
https://doi.org/10.3390/quantum5020022