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1. Acoustic metamaterials for realizing a scalable multiple phi-bit unitary transformation

   https://doi.org/10.1063/5.0188462  

   Runge, K.; Deymier, P_A; Hasan, M_A; Lata, T_D; Levine, J_A (February 2024, AIP Advances)

   The analogy between acoustic modes in nonlinear metamaterials and quantum computing platforms constituted of correlated two-level systems opens new frontiers in information science. We use an inductive procedure to demonstrate scalable initialization of and scalable unitary transformations on superpositions of states of multiple correlated logical phi-bits, classical nonlinear acoustic analog of qubits. A multiple phi-bit state representation as a complex vector in a high-dimensional, exponentially scaling Hilbert space is shown to correspond with the state of logical phi-bits represented in a low-dimensional linearly scaling physical space of an externally driven acoustic metamaterial. Manipulation of the phi-bits in the physical space enables the implementation of a non-trivial multiple phi-bit unitary transformation that scales exponentially. This scalable transformation operates in parallel on the components of the multiple phi-bit complex state vector, requiring only a single physical action on the metamaterial. This work demonstrates that acoustic metamaterials offer a viable path toward achieving massively parallel information processing capabilities that can challenge current quantum computing paradigms. 

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2. Practical implementation of a scalable discrete Fourier transform using logical phi-bits: nonlinear acoustic qubit analogues

   https://doi.org/10.1007/s40509-023-00312-5  

   Deymier, P A; Runge, K; Hasan, M A; Lata, T D; Levine, J A (December 2023, Quantum Studies: Mathematics and Foundations)

   It is shown that multiple logical phi-bit large-scale unitary operations analogous to quantum circuits can be realized by design. Logical phi-bits are nonlinear acoustic analogues of qubits which arise when elastic waveguides are coupled and driven at multiple frequencies in the presence of non-linearities. The contribution presents an approach that maps both the state of multiple phi-bits in their supporting nonlinear acoustic metastructure and their representations as complex state vectors in exponentially scaling Hilbert spaces. Upon physically actuating π changes in phi-bit phases and by engineering appropriate multiple phi-bits representations, one can realize a scalable phi-bit-based quantum Fourier transform. 

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3. Revolutionizing Computations: Quantum Circuit Analogues with Nonlinear Acoustic Waves

   Hasan, M Arif; Deymier, Pierre; Runge, Keith; Levine, Josh (March 2024, The American Institute of Physics, publisher, Bulletin of the American Physical Society)

   Quantum computing utilizes superposition and entanglement to surpass classical computer capabilities. Central to this are qubits and their use to realize parallel quantum algorithms through circuits of simple one or two qubit gates. Controlling and measuring quantum systems is challenging. Here, we introduce a paradigm utilizing logical phi-bits, classical analogues of qubits using nonlinear acoustic waves, supported by an externally driven acoustic metastructure. These phi-bits bridge a low-dimensional linearly scaling physical space to a high-dimensional exponentially scaling Hilbert space in which parallel processing of information can be realized in the form of unitary operations. Here, we show the implementation of a nontrivial three-phi-bit unitary operation analogous to a quantum circuit but achieved via a single action on the metastructure, whereby the qubit-based equivalent requires sequences of qubit gates. A phi-bit-based approach might offer advantages over quantum systems, especially in tasks requiring large complex unitary operations. This breakthrough hints at a fascinating intersection of classical and quantum worlds, potentially redefining computational paradigms by harnessing nonlinear classical mechanical systems in quantum-analogous manners, blending the best of both domains. 

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4. Demonstration of a two-bit controlled-NOT quantum-like gate using classical acoustic qubit-analogues

   https://doi.org/10.1038/s41598-022-18314-5  

   Runge, Keith; Hasan, M. Arif; Levine, Joshua A.; Deymier, Pierre A. (December 2022, Scientific Reports)

   Abstract The Controlled-NOT (CNOT) gate is the key to unlock the power of quantum computing as it is a fundamental component of a universal set of gates. We demonstrate the operation of a two-bit C-NOT quantum-like gate using classical qubit acoustic analogues, called herein logical phi-bits. The logical phi-bits are supported by an externally driven nonlinear acoustic metamaterial composed of a parallel array of three elastically coupled waveguides. A logical phi-bit has a two-state degree of freedom associated with the two independent relative phases of the acoustic wave in the three waveguides. A simple physical manipulation involving the detuning of the frequency of one of the external drivers is shown to operate on the complex vectors in the Hilbert space of pairs of logical phi-bits. This operation achieves a systematic and predictable C-NOT gate with unambiguously measurable input and output. The possibility of scaling the approach to more phi-bits is promising. 

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5. Realizing permutation gates with phi-bits: Acoustic quantum analogue computing

   https://doi.org/10.1063/5.0241680  

   Cavalluzzi, David; Ige, Akinsanmi_S; Runge, Keith; Deymier, Pierre_A (March 2025, Journal of Applied Physics)

   We present both the theoretical framework and experimental implementation of permutation gates using logical phi-bits, classical acoustic analogs of qubits. Logical phi-bits are nonlinear acoustic modes supported by externally driven acoustic metamaterials. Using a tensor product of modified Bloch sphere representations, we realize all possible two logical phi-bit permutations including SWAP and C-NOT. We also illustrate the scalability of a permutation for any number of logical phi-bits. Experimental demonstrations of these permutations require a single physical action on the driving conditions of the acoustic metamaterial. All logical phi-bits exist in the same physical system. We compare the phi-bit system with its quantum counterpart using Qiskit simulations, which illustrate the complexity of realizing these permutations in a quantum context. 

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