Realizing Shor’s algorithm with topological acoustic phase bits

Abstract

Quantum algorithms motivate alternative approaches to computation, and classical physical systems that generate correlations can enable parallelism. Here we present a framework for quantum-inspired computing based on phase bits (phibits), which represent logical units through the phases of nonlinear topological acoustic waves. We define two theoretical tools: the phase cache, which dynamically tracks the evolution of geometric phases, and the operator spectra shift, which enables consistent mapping between physical manipulations and computational operations. Using this framework, we implement the period finding core of Shor’s algorithm and demonstrate the factorization of composite integers 15 and 35. The experimental probability distributions for the measured outcomes show good agreement with theoretical predictions, validating the accuracy of the phibit implementation and the robustness of the nonlinear acoustic platform. These results show the potential of phibits as an approach to performing complex computational tasks.

Introduction

Quantum computing can provide computational advantages in certain tasks by exploiting nonclassical effects such as superposition and entanglement. These properties enable parallel processing, that can accelerate calculations in various fields, from cryptography and optimization to modeling complex physical systems^1,2,3,4,5. Despite considerable progress in qubit stability and coupling^6,7, the practical implementation of quantum computers is still limited by strict hardware and environmental requirements, including low-temperature operation, extreme noise isolation, and sophisticated error correction codes^5,8,9. These problems have stimulated the search for alternative approaches that reproduce some of the functional advantages of quantum information processing, while operating on more accessible classical platforms.

Topological acoustics (TA)^10,11 provides a framework for sound wave manipulating with controllable and reproducible phase responses. By revealing quantum-like properties in classical acoustic systems, such as geometric phases, enables coding and multiplexing strategies for information transmission and processing. Recent advances in TA have already demonstrated simple, low-cost, and highly configurable devices with potential for applications in computing, telecommunications, sensing, and information processing.

Based on these developments, the concept of an acoustic phase bit (phibit) was introduced as a classical analogue of a quantum bit (qubit). Unlike conventional qubits, phibits encode logical states in controlled geometric phases of nonlinear topological acoustic waves. Previous work^{12,13,14,15,16,17,18,19,20,21} has shown that phibits can reproduce the operational principles of qubit-based computing, including the realization of a universal set of gates¹² within the framework of a single mathematical model and with distinct physical manipulations. These results indicate that acoustic wave systems can serve not only as analogues of quantum circuits, but also as the basis for hybrid computational platforms.

In this manuscript, we extend the phibit framework by introducing two key theoretical contributions to mathematical representation of phibits. First, we develop the “phase cache”, a dynamic structure for tracking and managing the evolution of geometric phases, enabling precise control over multi-phibit computations. Second, we introduce the “operator spectra shift”, a method that guarantees consistent mappings between physical acoustic manipulations and computational operations by ensuring spectral disjointness in the Sylvester equation formalism. Using these tools, we implement the period finding core of Shor’s algorithm on a nonlinear acoustic platform. We experimentally demonstrate robust factorization of the composite integers N = 15 and N = 35, showing good agreement between theoretical predictions and measured probability distributions.

These results demonstrate an implementation of the quantum algorithm using a topological acoustics platform, illustrating how robust classical wave systems can implement key features of quantum computing, which directly corresponds to the goals of the current Nature collection on “Harnessing Topological Acoustics for Applied Solutions.”

Results

Shor’s algorithm overview

Shor’s algorithm^1,2 provides an exponential speedup for integer factorization by reducing it to the problem of order finding. The key idea is to determine the period r of the modular exponentiation function ${a}^{x}\,{{\rm{mod}}}\,\,N$, where N is the composite number to be factored and a is a randomly chosen integer such that 2 ≤ a < N. If the chosen value of a shares a non-trivial common factor with N, i.e., gcd(a, N) ≠ 1, a factor is immediately found without any further quantum processing. When gcd(a, N) = 1, the algorithm proceeds to determine the period r. Once r is known, the factors of N can be efficiently extracted by classically computing gcd(a^r/2 ± 1, N). This procedure succeeds only when r is even; if r is odd, the attempt fails and a different value of a must be selected.

The quantum portion of the algorithm focuses entirely on finding the period r. It uses two registers: a computational register and a period register. The computational register has size $n=\lceil {\log }_{2}(N)\rceil$ and stores the results of ${a}^{x}\,{{\rm{mod}}}\,\,N$. For example, when N = 15, the computational register requires n = 4 bits. The period register, which encodes the exponent x, generally requires about 2n bits in a standard implementation. Initially, the computational register is prepared in the state $\left|0\ldots 01\right\rangle$ and the period register in $\left|0\ldots 0\right\rangle$.

The algorithm begins by applying Hadamard gates to all bits in the period register, creating a uniform superposition over all possible values of the exponent

$$\frac{1}{\sqrt{{2}^{n}}}{\sum }_{x=0}^{{2}^{n}-1}\left|x\right\rangle ,$$

(1)

A controlled modular exponentiation is then performed, entangling the two registers so that the computational register holds the value of ${a}^{x}\,{{\rm{mod}}}\,\,N$ for each x in the superposition

$$\frac{1}{\sqrt{{2}^{n}}}{\sum }_{x=0}^{{2}^{n}-1}\left|x\right\rangle \left|{a}^{x}\,{{\rm{mod}}}\,\,N\right\rangle .$$

(2)

To extract the period, a phase estimation procedure is applied to the period register. In the original formulation of Shor’s algorithm, this is achieved using a full Quantum Fourier Transform (QFT) across the entire period register. The measurement of the transformed register yields information from which the period r can be deduced through classical post-processing techniques such as the continued fraction algorithm.

A direct implementation for factorizing N would therefore require a total of 3n bits: n for the computational register and 2n for the period register. However, Kitaev introduced²² a more efficient semiclassical variant of the QFT, known as Kitaev’s QFT (KQFT), which dramatically reduces the number of bits needed in the period register. In this approach, the 2n-bit register is replaced by a single bit that is repeatedly measured and reset up to 2n times in general. Classical feed-forward is used to apply conditional rotations based on previous measurement outcomes, allowing the algorithm to recycle the same bit throughout the procedure.

In standard quantum implementations of Shor’s algorithm, the controlled modular exponentiation step generally produces an entangled state between the exponent and computational registers. The algorithmic advantage is often described in terms of coherent superposition and interference enabled by this joint register structure. A complementary viewpoint is that quantum computation is an algorithmic use of Hilbert-space linearity and unitary transformations, and thus the formal state evolution, rather than the specific physical realization of microscopic quantum mechanics, is what matters at the computational level²³. In our platform, the computation is implemented on a classical system and therefore does not generate genuine quantum entanglement. Instead, strong classical nonlinear interactions between spectrally defined acoustic modes provide the physical mechanism that establishes interdependence (correlation) between the relevant degrees of freedom. In the phibit framework, these nonlinear mode correlations play the operational role of enabling the phase-based parallelism required by the period finding procedure, while remaining strictly classical in origin. A brief description of this nonlinear correlation mechanism is given in Supplementary Note 1.

Acoustic metastructure and geometric phases

Our experimental setup for implementing phibit operations is based on an acoustic metastructure consisting of three aluminum rods arranged in a linear array and glued together with epoxy. Each rod serves as an acoustic waveguide, while a piezoelectric transducer, coupled through a thin layer of honey, excites the system and ensures efficient transfer of vibrational energy. When the metastructure is driven by multiple sources, nonlinear interactions occur between the longitudinal acoustic modes of the rods. This process generates secondary waves at mixing frequencies f ⁽ⁱ⁾ = p⁽ⁱ⁾f₁ + q⁽ⁱ⁾f₂, where ${p}^{(i)},{q}^{(i)}\in {\mathbb{Z}}$ label the i-th phase channel (phibit). A detailed description of the experimental configuration is provided in Section Methods.

Topological acoustics (TA) is not limited to topological acoustic insulators with protected edge modes, but more generally covers situations where the topology of the parameter space or of the manifold spanned by state vectors plays a functional role, for example through geometric phases that encode the structure of that manifold¹¹. In our implementation the topology is mapped by the geometric phase structure associated with the multi-waveguide acoustic field. The collective acoustic response of this coupled system provides a framework for encoding and manipulating information through relative phase differences.

This work employs an open-loop geometric phase in real space. Operationally, the relevant quantities are the phase differences of the longitudinal displacement field between rods, evaluated at the waveguide termini and combined into phase differences across the array. Thus, the geometry here is the physical spatial arrangement of the waveguides and the relative displacement of the acoustic fields across them. For a given phibit i associated with mixing frequency f ⁽ⁱ⁾, the geometric phases are denoted by ${\varphi }_{12}^{(i)}$ and ${\varphi }_{13}^{(i)}$, where the subscripts specify which waveguides are being compared. The geometric phase along the open loop is explored parametrically through the driving frequencies.

Small static variations in rod alignment, epoxy bonding, or transducer coupling are part of the fixed experimental conditions. In practice, they are implicitly included in the measured geometric phases and do not by themselves generate logical errors provided they remain stable over the duration of an experimental run. The dominant error sources in this platform are instead dynamic phase fluctuations, which we quantify through the Monte Carlo robustness analysis.

The reproducibility and stability of the phibit platform have been established in our previous works^{10,12,14,15,16,17,18,19,20,21}. Experiments conducted over multiple weeks consistently show phase transitions with background phase fluctuations typically on the order of 10^∘ to 15^∘.

In principle, each distinct and resolvable mixing frequency f ⁽ⁱ⁾ within the usable bandwidth of the device can host a phibit. Thus, the total number of phibits is limited mainly by the range of detectable secondary frequencies and the requirement that their spectral lines remain sufficiently spaced for reliable phase extraction. Previous work has demonstrated that on the order of  ~ 50 such phase channels can be realized in a single device¹⁷ over a limited frequency range, indicating that the nine phibits used in the present experiment represent a conservative choice rather than a fundamental upper bound.

When considering the scalability of the system, the maximum usable number of phibits in a given device is set by several hardware parameters: the spectral properties of the metastructure (bandwidth and mode spacing), the precision and stability of the driving frequency adjustments, the number of rods (which determines how many independent waveguides are available for forming phase differences), and the number of independent control parameters. In the present implementation, only the drive frequency of a single rod is used as a tunable control parameter. In more complex metastructures with additional rods and drive channels, one could in principle vary not only the drive frequencies of multiple rods, but also their initial phases and amplitudes, either simultaneously or sequentially in time during the algorithm, or in tailored combinations to realize different computational steps. These additional degrees of freedom provide substantial potential for increasing the number of practically realizable phibits.

In this setting, a “physical manipulation” corresponds to changing one or both primary drive frequencies f₁ and/or f₂. Such a change shifts the phases of all phibits simultaneously because each f ⁽ⁱ⁾ depends on f₁ and f₂. By measuring, tracking, and controlling these phases, logical operations can be realized in analogy to qubit manipulations in a quantum system.

While the underlying hardware belongs to the broader class of topological acoustics, in the present work we do not engineer or exploit topologically protected propagation effects. Instead, we use the metastructure as a platform that provides a set of stable, controllable geometric phases and focus on the phibit representation, the phase cache, and the operator mapping frameworks that together enable the implementation of the period finding core of Shor’s algorithm on this acoustic system.

Bloch sphere representation and phase cache

In this framework, logical computational units are represented by phibits which are constructed from the experimentally measurable phases. Unlike quantum mechanical qubits, phibits provide a direct link between their abstract state and experimentally measurable quantities. Their parameters are explicitly constructed from combinations of phases from the known frequencies of the acoustic modes.

Each individual phibit i is represented by a two-component state vector

$$\left|{{{\mathcal{P}}}}_{i}\right\rangle ={e}^{i{\phi }_{2}^{(i)}}{\left(\sin {\phi }_{1}^{(i)},\cos {\phi }_{1}^{(i)}\right)}^{{\mathsf{T}}}=\alpha \left|0\right\rangle +\beta \left|1\right\rangle ,$$

(3)

where ${\phi }_{1}^{(i)}$ and ${\phi }_{2}^{(i)}$ are phase parameters uniquely determined by specific combinations of the underlying geometric phases ${\varphi }_{12}^{(i)}$ and ${\varphi }_{13}^{(i)}$ of the acoustic waves. These phase parameters define the complex amplitudes α and β, which describe the superposition of the logical basis states $\left|0\right\rangle$ and $\left|1\right\rangle$ of a phibit. Thus, by tuning the acoustic phases, one can precisely control the state of a phibit.

To track the temporal evolution of these parameters, we introduce a dynamic tool called the “phase cache,” which stores, for each phibit i, the accumulated phase information at computational step ξ:

$${\phi }_{j}^{(i)}(\xi )={g}_{j}^{(i)}\left({\varphi }_{12}^{(i)},{\varphi }_{13}^{(i)},\xi \right),\,j\in \{1,2\}.$$

(4)

Here ${g}_{j}^{(i)}$ is a design choice function that maps measured geometric phases to the Bloch-like parameters of a phibit. In the simplest case, g may select one geometric phase directly: g₁(φ₁₂, φ₁₃, ξ) = φ₁₂(ξ) and g₂(φ₁₂, φ₁₃, ξ) = φ₁₃(ξ). More generally, g can be any combination of the available geometric phases that gives a well-posed parametrization of the phibit state for the intended operations.

A related Bloch sphere representation has previously been shown to realize a universal set of gates¹². However, in that framework, multi-phibit operations impose simultaneous phase constraints on all participating phibits; even when only a single phibit is logically acted at a given step, the remaining phibits must either have no phase change or be back to their pre-operation values as shown previously.¹². To address this, we introduce the phase cache, which localizes updates to precisely those phibits involved in the current operation. Practically, this “disentangles” non-participating phibits.

A key distinction of the phibit architecture is that the primary control is global: changing the drive frequency modifies the geometric phase responses of all spectrally defined phibits in the metastructure. In our framework, the phase cache is precisely the tool that renormalizes this global physical action into an effective logical locality. The cache specifies which phase increments are algorithmically registered at each step. This logic is enabled by the nonlinear, multi-mode nature of the metastructure, where strong spectral coupling is intrinsic and can be used to implement multi-phibit operations.

This logic is illustrated schematically in Fig. 1. The upper panel shows the continuously varying physical geometric phases ${\varphi }_{12}^{(i)}$ for two example phibits under changes of the drive frequency f₁, while the lower panel shows the corresponding cached phases ${\phi }_{1}^{(i)}$ used for computation. This figure is provided as a conceptual visualization of the phase cache and is not based on experimental data.

Fig. 1: Schematic illustration of the phase cache for two hypothetical phibits.

Top: physical geometric phases ${\varphi }_{12}^{(i)}$ evolving with the applied driving frequency changes f₁. Bottom: cached Bloch sphere phases ${\phi }_{1}^{(i)}$ used in the algorithmic representation. Phibit i = 0 is shown by the solid black line and phibit i = 1 by the dashed orange line. The update sets are ${{{\mathcal{S}}}}^{(0)}=\{1,4\}$ and ${{{\mathcal{S}}}}^{(1)}=\{2,5\}$. The cached phases start from initial values ${\phi }_{1}^{(0)}(0)=\pi /2$ and ${\phi }_{1}^{(1)}(0)=-\pi /2$. The curves are not experimental data but illustrate the concept.

The description above treats ξ as a common logical step index for the full algorithm. In the experiment, phase updates are realized by continuously varying the drive frequency, starting from a chosen initial frequency and following a predefined frequency trajectory. Different phibits, and different Bloch sphere phases of the same phibit, can have distinct frequency intervals for the same logical step.

We therefore extend the phase cache to a two-layer structure consisting of: a logical phase cache that records which logical steps update each phibit; a physical phase cache that specifies, for each phibit i and each Bloch sphere phase j ∈ {1, 2}, the driving frequency segment associated with each logical step. We introduce a step-to-frequency map

$${{{\mathcal{F}}}}_{j}^{(i)}:\xi \,\mapsto \,\left({f}_{j,{{\rm{start}}}}^{(i)}(\xi ),{f}_{j,{{\rm{end}}}}^{(i)}(\xi )\right),$$

(5)

which partitions the experimental change into segments associated with logical step indices. In general, ${{{\mathcal{F}}}}_{1}^{(i)}\ne {{{\mathcal{F}}}}_{2}^{(i)}$ and ${{{\mathcal{F}}}}_{j}^{(i)}\ne {{{\mathcal{F}}}}_{j}^{(i{\prime} )}$, so that the same logical step index may correspond to different frequency ranges across phibits and across their phases.

Within this representation, a phibit’s cached phase changes only when there is an operation event on that phibit and between events it remains constant. The set of logical step indices at which phibit i undergoes a phase update is denoted by ${{{\mathcal{S}}}}^{(i)}$. To connect algorithmic phase updates to experimentally measured geometric phases, we introduce an affine calibration of the measured phase responses in the selected frequency band,

$${\widetilde{\varphi }}_{12}^{(i)}(f)={k}_{12}^{(i)}{\varphi }_{12}^{(i)}(f)+{b}_{12}^{(i)},\,{\widetilde{\varphi }}_{13}^{(i)}(f)={k}_{13}^{(i)}{\varphi }_{13}^{(i)}(f)+{b}_{13}^{(i)},$$

(6)

where ${k}_{12}^{(i)},{b}_{12}^{(i)}$ and ${k}_{13}^{(i)},{b}_{13}^{(i)}$ are obtained from constraints on the phase as function of drive frequency curves for each phibit. The algorithmic initial values ${\phi }_{j}^{(i)}(0)$ remain defined by design and are not used as physical offsets.

For each physical realization of a logical step $s\in {{{\mathcal{S}}}}^{(i)}$, we define the geometric phase increments using the frequency segments specified by ${{{\mathcal{F}}}}_{j}^{(i)}$ as

$$\Delta {\varphi }_{12}^{(i)}(s) = {\varphi }_{12}^{(i)}\left({f}_{1,{{\rm{end}}}}^{(i)}(s)\right)-{\varphi }_{12}^{(i)}\left({f}_{1,{{\rm{start}}}}^{(i)}(s)\right),\,\Delta {\varphi }_{13}^{(i)}(s) \\ = {\varphi }_{13}^{(i)}\left({f}_{2,{{\rm{end}}}}^{(i)}(s)\right)-{\varphi }_{13}^{(i)}\left({f}_{2,{{\rm{start}}}}^{(i)}(s)\right).$$

(7)

The affine transformation shifts do not contribute to the update rule,

$$\Delta {\widetilde{\varphi }}_{12}^{(i)}(s)={k}_{12}^{(i)}\Delta {\varphi }_{12}^{(i)}(s),\,\Delta {\widetilde{\varphi }}_{13}^{(i)}(s)={k}_{13}^{(i)}\Delta {\varphi }_{13}^{(i)}(s).$$

(8)

The evolution of the Bloch sphere phase parameters ${\phi }_{1}^{(i)}$ and ${\phi }_{2}^{(i)}$ for phibit i is defined as

$$ {\phi }_{1}^{(i)}(\xi )= {\phi }_{1}^{(i)}(0)+{\sum }_{\begin{array}{c}s\in {{{\mathcal{S}}}}^{(i)}\\ s\le \xi \end{array}}{k}_{12}^{(i)}\Delta {\varphi }_{12}^{(i)}(s),\,{\phi }_{2}^{(i)}(\xi )={\phi }_{2}^{(i)}(0) \\ \qquad\qquad+{\sum }_{\begin{array}{c}s\in {{{\mathcal{S}}}}^{(i)}\\ s\le \xi \end{array}}{k}_{13}^{(i)}\Delta {\varphi }_{13}^{(i)}(s),$$

(9)

where ${\phi }_{j}^{(i)}(0)$ are the algorithmically defined initial values and ${k}_{12}^{(i)}$, ${k}_{13}^{(i)}$ are sensitivity coefficients chosen by design. The terms $\Delta {\varphi }_{12}^{(i)}(s)$ and $\Delta {\varphi }_{13}^{(i)}(s)$ represent the calibrated geometric phase changes extracted from the physical manipulations associated with step s.

The phase cache thus serves as a structured record that links each phibit to its specific update history. For every phibit i, it maintains: the ordered set ${{{\mathcal{S}}}}^{(i)}$ of operation steps, the corresponding phase increments ${\{\Delta {\phi }_{j}^{(i)}(s)\}}_{s\in {{{\mathcal{S}}}}^{(i)}}$, and the associated physical frequency intervals ${\{{{{\mathcal{F}}}}_{j}^{(i)}(s)\}}_{s\in {{{\mathcal{S}}}}^{(i)}}$. At any step ξ, the current value of a phase parameter is reconstructed as the initial value plus the cumulative effect of all updates with indices s≤ξ. If no update occurs at ξ, the value remains unchanged, ensuring that phase changes are entirely event-driven in the logical description while being realized through frequency manipulations in the experiment. This can be expressed as

$${\phi }_{j}^{(i)}({\xi }^{+})=\left\{\begin{array}{ll}{\phi }_{j}^{(i)}({\xi }^{-})+\Delta {\phi }_{j}^{(i)}(\xi ), & \xi \in {{{\mathcal{S}}}}^{(i)},\\ {\phi }_{j}^{(i)}({\xi }^{-}), & \,{{\rm{otherwise}}}\,,\end{array}\right.$$

(10)

where ξ⁻ and ξ⁺ are steps immediately before and immediately after processing step ξ respectively.

Let n denote the number of phibits. The logical phase cache stores, for each phibit i, the update set ${{{\mathcal{S}}}}^{(i)}$ together with the phase increments for two channels. The physical cache additionally stores, for each (i, j) with j ∈ {1, 2}, the frequency interval associated with each update step. If $| {{{\mathcal{S}}}}^{(i)}|$ denotes the number of operation events involving phibit i, then the total number of cached update records scales as

$${{\mathcal{M}}} \sim {\sum }_{i=0}^{n-1}{\sum }_{j=1}^{2}| {{{\mathcal{S}}}}^{(i)}| =2{\sum }_{i=0}^{n-1}| {{{\mathcal{S}}}}^{(i)}| .$$

(11)

In the sparse regime, where each logical step involves only a small subset of phibits, $| {{{\mathcal{S}}}}^{(i)}|$ remains bounded or grows slowly with circuit depth, and the cache size increases approximately linearly with n. In the dense regime, where most steps involve most phibits, $| {{{\mathcal{S}}}}^{(i)}|$ becomes comparable to the total number of logical steps n_L, leading to ${{\mathcal{M}}}=O(n{n}_{{{\rm{L}}}})$.

The update rule in Eq. (10) is local in i, so the computational cost per logical step is proportional to the number of phibits addressed by that step.

Scalability limits and control resolution

The phase cache formalism in Eq. (10) makes explicit that, in our platform, a logical circuit is executed as a sequence of discrete, event-driven updates of the cached Bloch sphere parameters ${\phi }_{1}^{(i)}$ and ${\phi }_{2}^{(i)}$. Each update event $\xi \in {{{\mathcal{S}}}}^{(i)}$ is realized physically by driving through a designated control segment ${{{\mathcal{F}}}}_{j}^{(i)}(\xi )$ in the drive space, over which the experimentally measured phase response yields the required increment $\Delta {\phi }_{j}^{(i)}(\xi )$ (Supplementary Table 3 and Supplementary Fig. 3). In the present experiments the calibration was chosen so that these increments are π-like (approximately 180^∘), providing a convenient phase-step unit for counting physical resources.

To quantify the demanded number of phase updates, define an indicator

$${\chi }_{j}^{(i)}(\xi )=\left\{\begin{array}{ll}1, & \,{{\rm{if}}}\; {{\rm{the}}}\,j\,-{{\rm{th}}}\; {{\rm{Bloch}}}\; {{\rm{phase}}}\; {{\rm{of}}}\; {{\rm{phibit}}}\,{{\rm{i}}}\,{{\rm{is}}}\; {{\rm{updated}}}\; {{\rm{at}}}\; {{\rm{step}}}\,\xi ,\\ 0, & \,{{\rm{otherwise}}}\,,\end{array}\right.\, j \in \{1,2\}.$$

(12)

Then the total number of phase-update events required by phibit i over the full circuit is

$${{{\mathcal{J}}}}^{(i)}\equiv {\sum }_{\xi \in {{{\mathcal{S}}}}^{(i)}}{\sum }_{j=1}^{2}{\chi }_{j}^{(i)}(\xi ).$$

(13)

In the common case of this work, both ${\phi }_{1}^{(i)}$ and ${\phi }_{2}^{(i)}$ are updated whenever $\xi \in {{{\mathcal{S}}}}^{(i)}$, so ${{{\mathcal{J}}}}^{(i)}=2| {{{\mathcal{S}}}}^{(i)}|$. The circuit-level phase-step budget is therefore governed by the most heavily used phibits (largest $| {{{\mathcal{S}}}}^{(i)}|$), notably the reused period-register phibit in the KQFT procedure.

A second constraint is that each update $\Delta {\phi }_{j}^{(i)}(\xi )$ must be implemented on a control segment where the phase response is sufficiently smooth to support a predictable increment. Locally, we may approximate

$${\phi }_{j}^{(i)}({f}_{1})\approx {a}_{j}^{(i)}{f}_{1}+{b}_{j}^{(i)},\,{a}_{j}^{(i)}\approx \frac{{{\rm{d}}}{\phi }_{j}^{(i)}}{{{\rm{d}}}{f}_{1}},$$

(14)

so producing a π-like update requires a segment length

$$\Delta {f}_{j,\pi }^{(i)}\approx \frac{\pi }{| {a}_{j}^{(i)}| }.$$

(15)

If the usable control window is ${f}_{1}\in [{f}_{\min },{f}_{\max }]$ with width ${W}_{f}={f}_{\max }-{f}_{\min }$, then the maximum number of disjoint π-like update segments available to component (i, j) is bounded by

$${K}_{j}^{(i)}\lesssim \frac{{W}_{f}}{\Delta {f}_{j,\pi }^{(i)}}\approx \frac{{W}_{f}| {a}_{j}^{(i)}| }{\pi },$$

(16)

up to the requirement that the specific segments align with the logical ordering defined by the phase cache. A necessary feasibility condition for executing a circuit on fixed hardware is therefore

$${{{\mathcal{J}}}}^{(i)}\le {\sum }_{j=1}^{2}{K}_{j}^{(i)}\,\,{{\rm{for}}}\; {{\rm{all}}}\; {{\rm{phibits}}}i{{\rm{used}}}\; {{\rm{by}}}\; {{\rm{the}}}\; {{\rm{circuit}}}\,,$$

(17)

with the tightest constraint typically coming from ${\max }_{i}{{{\mathcal{J}}}}^{(i)}$.

Let ε_ϕ denote the allowed phase error per update, set either by a desired gate-level tolerance or empirically by robustness analysis (as in the Monte Carlo study). For a drive increment uncertainty δf₁ (including discretization and drift) and measurement noise ${\eta }_{j}^{(i)}$, a first-order propagation model gives

$$| \delta {\phi }_{j}^{(i)}| \approx | {a}_{j}^{(i)}| | \delta {f}_{1}| +| {\eta }_{j}^{(i)}| \lesssim {\varepsilon }_{\phi },$$

(18)

hence the control-resolution requirement

$$| \delta {f}_{1}| \lesssim \frac{{\varepsilon }_{\phi }-| {\eta }_{j}^{(i)}| }{| {a}_{j}^{(i)}| },$$

(19)

whenever the right-hand side is positive. Equations (15)–(19) highlight the hardware tradeoff: steeper phase response $| {a}_{j}^{(i)}|$ increases the step density (16) but also tightens the required frequency precision (19). Conversely, flatter response relaxes precision but reduces the number of available steps within a fixed W_f.

The present implementation uses a single global control parameter (a sweep of f₁ with fixed f₂), so all phibits must share one trajectory and the feasible depth is limited by the most constrained phibit(s) in (17). More generally, let ${{\boldsymbol{\theta }}}\in {{\mathbb{R}}}^{d}$ denote a vector of independent controls (e.g., multiple drive frequencies, drive phases, amplitudes, or independently driven rods). Then, locally,

$$\delta {\phi }_{j}^{(i)}\approx {\nabla }_{{{\boldsymbol{\theta }}}}{\phi }_{j}^{(i)}\cdot \delta {{\boldsymbol{\theta }}},$$

(20)

so increasing d provides two direct scalability benefits: precision sharing, where the same ε_ϕ can be met with coarser resolution per control parameter if ∇_θϕ is distributed across components; and step-budget expansion, because updates can be multiplexed in time across different control directions (e.g., alternating sweeps in different drives) or implemented along longer, piecewise trajectories in a higher-dimensional control space, increasing the number of usable segments beyond the single-parameter bound (16).

The number of rods R in the metastructure impacts both observability and control. At each mixing frequency channel, choosing one rod as a reference gives up to (R − 1) independent phase differences, enabling richer design choices for the mapping functions ${g}_{j}^{(i)}$ (and thus potentially more robust phibit parametrizations), and redundant phase readouts that can reduce the effective ${\eta }_{j}^{(i)}$ in (18) via averaging. In addition, adding rods naturally enables additional independent drive channels (increasing d), which, by (20), directly enlarges the feasible phase-step budget.

In summary, within the phase cache architecture, physical scalability is governed by: the circuit-imposed update counts ${{{\mathcal{J}}}}^{(i)}$; the number of reliable close to linear control segments ${K}_{j}^{(i)}$ available within the usable control window; the per-update tolerance ε_ϕ as constrained by control resolution and phase noise. Increasing the number of independent controls and/or rods provides a systematic route to increasing feasible circuit depth by expanding the phase-step budget and improving robustness.

Mapping logical gates to physical manipulations

We now introduce a general representation for mapping phibit states to physical manipulations. Let us introduce the representation transform X which acts on the fundamental phibit state $\left|{{{\mathcal{P}}}}_{i}\right\rangle$ to produce the operational state

$$\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle =X\left|{{{\mathcal{P}}}}_{i}\right\rangle ,$$

(21)

When a physical process induces phase shifts, the state evolves according to

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{{{\rm{phys}}}}=X{\left|{{{\mathcal{P}}}}_{i}\right\rangle }_{{{\rm{phys}}}}=XM\left|{{{\mathcal{P}}}}_{i}\right\rangle \,\,{{\rm{with}}}\,\,{\left|{{{\mathcal{P}}}}_{i}\right\rangle }_{{{\rm{phys}}}}=M\left|{{{\mathcal{P}}}}_{i}\right\rangle ,$$

(22)

where M is the transform describing the physical manipulation on phibit state $\left|{{{\mathcal{P}}}}_{i}\right\rangle$. Following from Eq. (3), phase changes Δϕ₁ and Δϕ₂ lead to state evolution that can be decribed, for an n-phibit state, by

$$M(\Delta {\phi }_{1},\Delta {\phi }_{2})={e}^{i\Delta {\phi }_{2}}{\otimes }_{k=1}^{n}{{\mathcal{R}}}\left(\Delta {\phi }_{1}\right),\,{{\mathcal{R}}}(\theta )=\left(\begin{array}{ll}\cos \theta & \sin \theta \\ -\sin \theta & \cos \theta \end{array}\right).$$

(23)

On the other hand, when a logical gate U_gate acts on a phibit, we have

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{{{\rm{gate}}}}={U}_{{{\rm{gate}}}}\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle ={U}_{{{\rm{gate}}}}X\left|{{{\mathcal{P}}}}_{i}\right\rangle .$$

(24)

To ensure physical and logical descriptions are equivalent, we require

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{{{\rm{phys}}}}={\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{{{\rm{gate}}}}$$

(25)

for all realizations of the initial state $\left|{{{\mathcal{P}}}}_{i}\right\rangle$. Thus, for any logical input encoded by the measured geometric phases (via Eq. (3) and the phase cache), the experimentally applied frequency manipulation that produces the phase increments Δϕ_1,2 must induce, after representation by X, the same transformation of the operational state as the intended logical gate U_gate.

Operationally, the physical operator M is not an abstract assumption: it is constructed from the experimentally measured phase changes associated with the frequency segments (Eq. (5)). The update sets ${{{\mathcal{S}}}}^{(i)}$ specify which phibits are affected at each logical step, and the corresponding $\Delta {\phi }_{1,2}^{(i)}$ are extracted. Thus, M encodes the experimentally driven action of a given physical step, while U_gate encodes the target logical action of that step.

Imposing Eq. (25) for all input states implies

$${U}_{{{\rm{gate}}}}X-XM=0,$$

(26)

which expresses that, in the operational representation, the experimentally realized manipulation M acts as the desired logical gate U_gate. Equation (26) is a homogeneous Sylvester equation^24,25,26 and therefore always admits the trivial solution X = 0. Moreover, when nontrivial solutions exist they are generally not unique.

To select a unique, well conditioned representation transform, we introduce a modification by solving instead the inhomogeneous Sylvester equation

$${U}_{{{\rm{gate}}}}X-X\overline{M}=I,$$

(27)

where I is the identity matrix and $\overline{M}$ is a shifted version of the physical operator defined below. The nonzero right-hand side removes the trivial solution and, together with a spectral disjointness condition, guarantees existence and uniqueness of solution X.

Operator spectral shift for solvability

A standard result for Sylvester equations states that the inhomogeneous equation (27) has a unique invertible solution provided that the spectra of U_gate and $\overline{M}$ are disjoint. We enforce this by applying an operator spectral shift to the physical transform,

$$\overline{M}=M+\gamma I,$$

(28)

where γ is chosen such that

$$\sigma ({U}_{{{\rm{gate}}}})\cap \sigma (\overline{M})=\varnothing .$$

(29)

With X obtained from (27), the operational mapping associated with the physical manipulation can be written as

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{{{\rm{gate}}}}={U}_{{{\rm{gate}}}}\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle =(X\overline{M}+I)\left|{{{\mathcal{P}}}}_{i}\right\rangle ,$$

(30)

with $\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle =X\left|{{{\mathcal{P}}}}_{i}\right\rangle$. Since X is nonsingular, we can recover the corresponding phibit state by $\left|{{{\mathcal{P}}}}_{i}\right\rangle ={X}^{-1}\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle$.

Multi-stage Sylvester mapping

We now generalize the framework to handle an arbitrary sequence of quantum-like operations, such as those in Shor’s algorithm for different choices of N and base a. This includes multiple modular exponentiation blocks, controlled rotations between period-register phibits, and final measurement steps. Each stage of the algorithm is represented by a computational gate operator U_s, where the index s = 1, …, L enumerates the operations in time order. The gate-level logical operators that constitute U_s in the circuits of Figs. 2–4 are defined in Supplementary Note 3.

Fig. 2: Schematic of the period finding circuit used in Shor’s algorithm.

The upper wires represent the period finding register, initialized in $\left|+\right\rangle$ and processed one control phibit at a time: each control phibit (top) conditionally applies a modular exponentiation to the computational register (bottom), followed by single-phibit rotations (here R_π/2 and R_π/4) and a Hadamard gate H before measurement. Meter symbols denote measurements of the control phibits; the resulting classical outcomes are carried forward on the classical (double) wires to implement the required feed-forward conditioning in subsequent steps. The lower bundled wire indicates an n-phibit computational register, initialized as $\left|{{{\mathcal{P}}}}_{{{\rm{Init}}}}\right\rangle$, on which the controlled modular exponentiation blocks (Mod Exp) act: ${U}_{a,N}^{4}$ (red; Mod Exp A), ${U}_{a,N}^{2}$ (green; Mod Exp B), and ${U}_{a,N}^{1}$ (blue; Mod Exp C), corresponding to successive powers of the base a modular multiplication modulo N. Filled control dots and connecting vertical wires indicate controlled operations between the control phibit and the computational register. Blue dashed vertical markers (step 1 -- step 9) indicate the sequential stages of the procedure.

For each step s, we introduce a representation matrix X_s and a shifted physical manipulation operator

$${\overline{M}}_{s}={M}_{s}+{\gamma }_{s}I,$$

(31)

where γ_s is a real scalar shift parameter and I is the identity matrix of the full Hilbert space. The representation transform X_s is found from the Sylvester equation

$${U}_{s}{X}_{s}-{X}_{s}{\overline{M}}_{s}=I.$$

(32)

This equation ensures that the computational action U_s at step s can be reproduced using the physical manipulation ${\overline{M}}_{s}$. The solution X_s encodes the logical-to-physical correspondence for that step.

We propagate through each step by first computing the preimage

$${\left|{{{\mathcal{P}}}}_{i}\right\rangle }_{s}={X}_{s}^{-1}{\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{s-1},$$

(33)

where ${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{s-1}$ is the operational state exiting the previous step, with ${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{0}$ being initial state. The mapped output of step s is then

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{s}={X}_{s}{\overline{M}}_{s}{\left|{{{\mathcal{P}}}}_{i}\right\rangle }_{s}+{\left|{{{\mathcal{P}}}}_{i}\right\rangle }_{s},$$

(34)

which by construction equals the application of U_s to the operational state as

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{s}={U}_{s}{\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{s-1}.$$

(35)

By iterating this procedure for all L steps, the overall mapping reproduces the full computation is

$${U}_{{{\rm{chain}}}}={U}_{L}{U}_{L-1}\cdots {U}_{2}{U}_{1},$$

(36)

with the final ideal state

$${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{L}={U}_{{{\rm{chain}}}}\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle .$$

(37)

Algorithm execution

Several experimental realizations of Shor’s algorithm have been implemented across different quantum hardware platforms^27,28,29,30. The earliest demonstration²⁷ was later shown to be so oversimplified that it was effectively equivalent to coin flipping³¹. While some subsequent implementations adopted similar simplifications, others executed non-trivial instances of the algorithm. Factorization of N = 15 and N = 21 has been demonstrated experimentally, and an attempt to factor N = 35 was reported but unsuccessful³⁰.

Here we extend these efforts using our acoustic phibit platform. First, we factor N = 15 for bases a ∈ {2, 7, 8, 11, 13} to demonstrate the generality of our framework across distinct modular exponentiation realizations. Then we factor N = 35 with the non-trivial base a = 4, for which we are not aware of a reported successful experimental implementation of Shor’s algorithm on quantum hardware.

Following the approaches of^29,30, we implement the period finding core using the circuit shown on Fig. 2. The three modular exponentiation stages correspond to ${U}_{a,N}^{4}$, ${U}_{a,N}^{2}$, and ${U}_{a,N}^{1}$, each controlled by a distinct phibit in the period register; after each stage, a KQFT controlled rotations and Hadamard gate is applied. The computational register carries the modular arithmetic, while the period register accumulates phase information about the order r. The logical definitions of the single- and multi-phibit gates used throughout these circuits (Hadamard, phase rotations, controlled-U, CNOT, SWAP, and CSWAP) are summarized in Supplementary Note 3.

The modular exponentiation operator is defined as

$${U}_{a,N}^{k}:\,\left|x\right\rangle \,\mapsto \,\left|{a}^{k}x\,{{\rm{mod}}}\,\,N\right\rangle ,$$

(38)

with concrete circuit realizations shown for N = 15 and N = 35 in Figs. 3 and 4, respectively (see Supplementary Note 3 for the gate-level operator definitions). Each entire modular exponentiation block is controlled by its corresponding period-register phibit.

Fig. 3: Example modular exponentiation block circuits for factoring N = 15 with bases a ∈ {2, 7, 8, 11, 13}.

Here ${U}_{a,15}^{k}$ denotes modular multiplication by a^k modulo 15 acting on the computational register. Panels show the three controlled stages used in the period finding routine: (a) the ${U}_{a,15}^{4}$ block (red; executed at step 2), (b) the ${U}_{a,15}^{2}$ block (green; step 5; identical for the shown bases except the special case a = 11), and (c)-(g) the ${U}_{a,15}^{1}$ block (blue; decomposed into substeps 8.1--8.4 depending on a), shown explicitly for a = 2, 7, 8, 11, 13. Horizontal wires labeled $\left|{{{\mathcal{P}}}}_{j}\right\rangle$ denote the computational register basis states (here j = 0, 1, 2, 3); each block is applied conditionally, controlled by the corresponding phibit from the period finding register. Dashed vertical markers label the internal substep sequencing within a block realization.

Fig. 4: Example modular exponentiation block circuits for factoring N = 35 with base a = 4.

Here ${U}_{4,35}^{k}$ denotes modular multiplication by 4^k modulo 35 acting on the computational register. Shown for the three controlled stages used by the period finding routine: (a) ${U}_{4,35}^{4}$ (red; decomposed into substeps 2.1--2.2), (b) ${U}_{4,35}^{2}$ (green; substeps 5.1--5.5), and (c) ${U}_{4,35}^{1}$ (blue; substeps 8.1--8.7). Horizontal wires labeled $\left|{{{\mathcal{P}}}}_{j}\right\rangle$ denote the computational register basis states (here j = 0, …, 5); each block is applied conditionally, controlled by the corresponding phibit from the period finding register. Dashed vertical markers label the internal substep sequencing within a block realization.

We note that in this classical phibit platform gate stability should not be interpreted like for the conventional quantum computers maintaining coherent unitary operations against decoherence. The relevant limitation is classical noise in the experimentally measured and applied geometric / Bloch sphere phases. We therefore assess stability through the robustness of the phase updates to bounded phase perturbations across the full algorithm.

Measured outcome probabilities are extracted from the coefficients of the final operational state ${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{L}$ after all physical manipulations. In our experiments, phase noise up to  ~ 20^∘ was observed in the geometric phases as reported in the literature.¹² The Bloch sphere phase deviations extracted from the experiment reach up to  ~ 70^∘, as reported in Supplementary Table 3. To assess robustness, we therefore performed a Monte Carlo study in which, at every computational step, independent phase perturbations were sampled uniformly from [ − 75^∘, 75^∘] and applied separately to the two Bloch sphere parameters in the phibit representation defined by Eq. (3). Deviations from the ideal theoretical distribution are quantified using the squared statistical overlap (SSO). The worst-case scenario probability distributions obtained from the Monte Carlo simulations are shown in Fig. 5. We note that these deviations can be reduced by choosing a finer frequency step grid for f₁, however, the chosen 50Hz step already demonstrates a high degree of robustness for the current study.

Fig. 5: Theoretical output probability distributions (black) compared with the worst-case distributions observed over Monte Carlo experiments (orange), illustrating the effect of geometric phase noise on the measured period-register outcomes (decimal 0–7).

In each run, a phibit circuit is evaluated through successive stages, and at every stage two independent phase increments (δϕ₁, δϕ₂) are sampled uniformly from $[-{\phi }_{\max },{\phi }_{\max }]$ with ${\phi }_{\max }=1.35\,{{\rm{rad}}}$; the similarity to the ideal distribution is quantified by the squared statistical overlap (SSO). For each parameter set, ${n}_{\exp }=100$ Monte Carlo experiments were performed and the plotted worst-case distribution corresponds to the run with minimal SSO. Panel details: a N = 15, a = 2, r = 4, SSO = 0.997514; b N = 15, a = 7, r = 4, SSO = 0.997548; c N = 15, a = 8, r = 4, SSO = 0.997046; d N = 15, a = 11, r = 2, SSO = 0.998430; e N = 15, a = 13, r = 4, SSO = 0.997242; f N = 35, a = 4, r = 6, SSO = 0.999599. Here r denotes the period of the modular exponentiation defined by the corresponding number to be factorized N and base a.

Unlike in classical quantum computer implementations where the output distribution is gathered from repeated measurements over many experiments, in this framework a single experimental run of the full physical manipulation protocol is sufficient to reconstruct the entire probability distribution. This is because, within our Sylvester equation representation, the final operational state ${\left|{\widetilde{{{\mathcal{P}}}}}_{i}\right\rangle }_{L}$ is obtained from the experimentally measured phase increments and the corresponding stepwise mappings X_s, giving direct access to the state coefficients from which the outcome probabilities are computed. Details of the Monte Carlo simulations, SSO metric, and the performance statistics across runs are provided in Supplementary Note 2, including histograms of SSO over all trials (Supplementary Fig. 1) and the Monte Carlo mean output distributions with uncertainty quantification relative to the ideal probabilities (Supplementary Fig. 2).

Discussion

We have demonstrated that phibits in a nonlinear topological acoustic metastructure can execute quantum-like computations by encoding logical state in geometric phases of the acoustic field. We introduced two key tools: the phase cache, which dynamically track phase updates across phibits, and the operator spectra shift, which guarantees a well-posed Sylvester equation in finding mappings between desired gate operations and physical manipulations.

Using these tools within a phibit framework, we implemented the period finding core of Shor’s algorithm factoring N = 15 and N = 35 integers. The experimental and theoretical probability distributions show strong agreement. The worst-case examples from Monte Carlo simulations and theoretical predictions closely align in relative weights. Phibits thus define a physically grounded, phase encoded computing platform in which logical states and gate operations are built from experimentally measurable geometric phases of nonlinear acoustic modes.

From a technological perspective, phibits offer several practical advantages relative to competing quantum hardware approaches. First, they operate at room temperature on a classical platform, avoiding the cryogenic and stringent environmental requirements typical of superconducting qubits. Second, the experimental implementation relies on standard and inexpensive acoustic components, making the setup accessible and highly configurable. Third, phibits encode information in directly measurable relative phases of acoustic fields. These properties position phibits as an alternative way for exploring quantum-inspired algorithms, whereas conventional quantum hardware remains costly or operationally demanding.

The use of multiple phibits within the same waveguide array does not, by itself, introduce direct logical cross-talk between phibits. All phibits reside in the same physical metastructure and correspond to different vibrational modes of the same acoustic system, they are defined spectrally: each phibit is associated with a distinct mixing frequency and is read out via phase measurements. As long as the mixing frequencies remain spectrally resolved, the logical phibits remain decoupled at the level of the representation. Increasing the number of phibits does not introduce a new mechanism of logical cross-talk.

In the present work, the acoustic platform is macroscopic, but the logical architecture does not rely on this specific form factor. In this framework, the scalability does not directly depend on the spatial dimensions of the device or on physical miniaturization, but rather on the number of resolvable mixing frequencies and the quality of the available control parameters.

A key scalability distinction should be emphasized relative to analog signal-based emulation approaches that encode an entire n-qubit state within a single multi-tone classical waveform. In such schemes, the number of distinct spectral components needed to represent basis amplitudes grows rapidly with n, so bandwidth and time-resolution constraints translate into increasingly strict spacing and filtering requirements, motivating practical limits on the order of 40–50 qubits³². In contrast, our architecture does not attempt to embed an exponentially large state into one frequency component: each phibit is defined on a resolvable nonlinear mixing frequency line and accessed through directly measured geometric phase differences. Consequently, the number of occupied spectral lines scales linearly with the number of phibits rather than exponentially with the Hilbert-space dimension. The phase cache also scales in a linear way, as discussed in the Results.

In terms of entanglement, our platform is entirely classical and therefore does not generate genuine quantum entanglement. Instead, the relevant resource is the strong nonlinear correlation between spectrally defined acoustic modes, described as classical entanglement in the context of acoustic qubit analogues²¹. Within the phibit framework, this mode correlation, together with the phase cache and the multi-stage Sylvester mapping with operator spectra shift, provides the effective quantum-like parallelism needed to implement the period finding core of Shor’s algorithm and to reproduce the expected outcome distributions. Thus, we do not rely on maintaining quantum coherence or entanglement in the system, but rather on controlled, measurable geometric phase relations across coupled nonlinear channels.

One needs to differentiate between correlation that serves as a resource for realizing parallelism in phase-based computing and the mechanisms underlying correlation. In a multipartite system, correlation establishes an interdependency among relevant degrees of freedom; in quantum systems, entanglement is a uniquely quantum mechanism that can generate correlations with no classical counterpart. In the present work, however, the interdependency between degrees of freedom arises entirely from classical nonlinear dynamics of coupled acoustic modes. Specifically, the correlation we discuss is produced by nonlinear coupling within a driven array of coupled waveguides whose dynamics can be mapped to interacting nonlinear oscillators, as introduced and analyzed in Refs. ^12,16. In this classical setting, correlation means that the measured outputs of different modes are not statistically independent. They are linked through a shared nonlinear interaction, so the evolution of one mode depends on the state of the others, producing correlated outcomes after the interaction. Thus, the nonlinear correlation between phibits enables parallelism in phibit computing, without invoking quantum nonlocality or entanglement; a brief mathematical description is provided in Supplementary Note 1.

These results establish a phibits framework as a robust approach to implement quantum-like algorithms, while operating at room temperature on a classical system. Current limitations arise from phase noise and finite control resolution.

Methods

Experimental setup

Our experimental setup is constructed from individual aluminum rods (McMaster-Carr part number 1615T172), each possessing a 1/2 inch diameter and a length of 0.6096 m. The material’s density is 2660 kg/m³. These rods are arranged in a linear configuration, with epoxy (50176 KwikWeld Syringe) filling the lateral gaps. This epoxy serves a dual purpose: providing structural rigidity and facilitating efficient acoustic wave propagation between the rods. The longitudinal acoustic waves induced within these waveguides are tuned to an approximate wavelength of 10cm, corresponding to the intended operating frequency.

Dynamic excitation of the waveguides is achieved through ultrasonic longitudinal contact transducers (Olympus IMS V133-RM) precisely affixed to the distal ends of each rod. On the transmission side, these transducers are driven by discrete B&K Precision 4055B signal generators in conjunction with PD200 amplifiers. Conversely, each detecting transducer is interfaced with a dedicated channel of a Tektronix MDO3024 oscilloscope, enabling signal acquisition and temporal data extraction across the entire array.

Optimal energy transfer from the transducers to the rods is realized through the application of a thin layer of commercial honey as a coupling medium. Empirical testing demonstrated honey performed well in our tests for transmitting longitudinal vibrational energy within our operational frequency range. To maintain consistent transducer contact pressure, elastic rubber bands (Walmart, Cat. No. 564755837) are employed. The entire array is suspended by threads to minimize external vibrational interference. This suspension provides sufficient isolation for stable phase measurements within the working frequency range used in this study.

The experiments were conducted in an air-conditioned laboratory environment with a typical temperature range of approximately 18 ^∘C to 22 ^∘C. Within this range, we did not observe a measurable impact on the phase measurements. Long-term reproducibility of the phase behavior has also been established in our prior studies across multiple experiments^{10,12,14,15,16,17,18,19,20,21}.

Phase cache

The phase cache specifies when phibit’s phases are updated during a computation. For every phibit $\left|{{{\mathcal{P}}}}_{i}\right\rangle$, we define a set of update indices ${{{\mathcal{S}}}}^{(i)}$, whose elements may include substeps (e.g., k. m), at which the corresponding phibit phase is updated. These steps are depicted in Fig. 2, while concrete substep realizations are shown in Figs. 3 and 4. In accordance with Eq. (10), every index $\xi \in {{{\mathcal{S}}}}^{(i)}$ triggers a phase update for phibit $\left|{{{\mathcal{P}}}}_{i}\right\rangle$ on the relevant phase components, and the Bloch sphere parameters ${\phi }_{j}^{(i)}(\xi )$ are updated accordingly.

For all circuit realizations considered, the sets ${{{\mathcal{S}}}}^{(i)}$ are given in Table 1 with respect to a circuit realization (N, a) and phibit index i ∈ {0, …, 6}. The logical period register that controls the modular exponentiation gates ${U}_{a,N}^{1}$, ${U}_{a,N}^{2}$, and ${U}_{a,N}^{4}$ is implemented using a single reused phibit. Specifically, for the (15, a) realizations this role is carried by $\left|{{{\mathcal{P}}}}_{4}\right\rangle$, while for (35, 4) it is carried by $\left|{{{\mathcal{P}}}}_{6}\right\rangle$. The required phase increments are therefore applied only at the update indices listed in Table 1 for the corresponding (N, a) realization, with all other non-participating phibits left unchanged at those steps.

Table 1 Phase cache map

Physical manipulation protocol

This section summarizes the experimentally implemented parameter values and the resulting step structure used to realize the algorithm.

Phibits are defined by integer pairs (p⁽ⁱ⁾, q⁽ⁱ⁾) and their corresponding frequency f ⁽ⁱ⁾ = p⁽ⁱ⁾f₁ + q⁽ⁱ⁾f₂. The specific linear combination choices and the associated affine calibration parameters $\left({k}_{12}^{(i)},{b}_{12}^{(i)}\right)$ and $\left({k}_{13}^{(i)},{b}_{13}^{(i)}\right)$ used in this work are listed in Table 2. These coefficients are substituted into Eq. (6) to convert the measured geometric phases to the calibrated increments used for logical updates.

Table 2 Phibit linear combination and affine calibration parameters

Operationally, each physical step is implemented by updating the drive frequency f₁ by Δf = 50 Hz using the signal generator while keeping f₂ fixed. After each update, the system is allowed to reach a steady response before acquiring time-domain signals at the receiving transducers. The geometric phases ${\varphi }_{12}^{(i)}$ and ${\varphi }_{13}^{(i)}$ are extracted from the measured longitudinal displacement fields at the rod termini by comparing the complex phases of the corresponding spectral components at the mixing frequency f ⁽ⁱ⁾. These measured phase differences define the experimentally accessible inputs for the affine calibration in Eq. (6) and, through the phase cache update rule Eq. (10), determine the applied Bloch sphere phase increments used to implement the logical steps of the algorithm.

For the algorithm implementation we held f₂ fixed at 90kHz and varied f₁ in discrete 50Hz increments, chosen for experimental convenience; a finer frequency grid could be used if needed. The experimental geometric phases ${\varphi }_{12}^{(i)}$ and ${\varphi }_{13}^{(i)}$ and the selected frequency window used for computation are shown in Supplementary Fig. 4. The protocol starts at f₁ = 71kHz and increases to f₁ = 73kHz. Over this interval, each Bloch sphere phase is chosen to have approximately twelve successive changes of about 180^∘, which defines our twelve physical steps. The affine calibration parameters in Table 2 were chosen so that these phase responses provide the required number of resolvable physical steps within the selected frequency window.

The resulting Bloch sphere phase changes are shown in Supplementary Fig. 3. The corresponding physical phase cache is reported in Supplementary Table 3, which list, for each phibit and for each Bloch sphere phase, frequency intervals $\left({f}_{{{\rm{start}}}},{f}_{{{\rm{end}}}}\right)$, the frequency increment Δf, the Bloch sphere phase increment Δϕ, and the step deviation defined as Δϕ − 180^∘. Additionally, the step based phase cache in Supplementary Table 4 summarizes the participating phibits at each logical step.

The phase increments listed in Supplementary Tables 5 and 7 are those required by the multi-stage Sylvester mapping for the logical operators in the corresponding circuit, and the frequency intervals in Supplementary Table 3 provide their direct experimental realization.

For the reference circuit (N, a) = (15, 2), with the remaining (15, a) realizations constructed in the same manner, the applied Bloch sphere phase increments $\{\Delta {\phi }_{1}^{(i)}\}$ and $\{\Delta {\phi }_{2}^{(i)}\}$ assigned to each logical step are summarized in Supplementary Table 5. The resulting phase state evolution, expressed as the accumulated values $\{{\phi }_{1}^{(i)}\}$ and $\{{\phi }_{2}^{(i)}\}$ together with the readouts of the reused period register phibit, is reported in Supplementary Table 6. For (N, a) = (35, 4), the analogous applied increments and phase state evolution are given in Supplementary Tables 7 and 8, respectively. The final measured phase pair is mapped to a logical measurement outcome.

In this platform, the physical realization of logical states and operations differs from the architecture implied by standard quantum hardware intuition. All phibits coexist within the same nonlinear topological metastructure and are addressed through global drive conditions and controlled frequency adjustments, rather than through pairwise wiring or spatially separated control lines. As a result, multi-phibit operations do not rely on a dedicated physical interconnect that maps one phibit as a control and another as a target. Instead, the effective logical action comes from the nonlinear coupling and the phase cache update rules, which prescribe which phibits have a phase increment at each computational step. Because the system is classical, the relevant errors are classical phase and frequency fluctuations.

Robustness analysis under phase fluctuations

In conventional qubit-based hardware, crosstalk typically refers to unintended interactions between qubits or leakage of control fields: an operation addressed to one qubit can induce rotations or dephasing on other qubits through residual couplings, shared control lines, or imperfect isolation. In the present phibit platform, the notion of crosstalk is structurally different. All phibits coexist in the same metastructure and are driven by the same global control conditions, while each phibit is defined spectrally (by a distinct nonlinear mixing frequency) and accessed through phase readout at that frequency. As a result, there is no notion of hardware-level phibit-to-phibit crosstalk; instead, the relevant coupling is control-to-phase cross-sensitivity (global drive changes affecting multiple phase channels).

Crucially, the phase cache converts this globally driven physics into an event-driven logical update rule. At each logical step ξ, only a prescribed subset of phibits participates (Supplementary Table 4), and only their cached Bloch sphere parameters are updated according to Eq. (10). Phibits that do not participate at step ξ are not registered by the cache and therefore cannot induce logical errors at that step. In this sense, the phase cache suppresses logical crosstalk with the rest of the register by construction: only the phibits involved in the current computational step contribute to the algorithmic state evolution.

We model phase imperfections at the level of the cached phase increments. Let ${{\boldsymbol{\theta }}}\in {{\mathbb{R}}}^{d}$ denote the vector of experimental controls and let ${{\boldsymbol{\phi }}}\in {{\mathbb{R}}}^{m}$ collect the Bloch sphere phase parameters used for computation (two per phibit in the representation of Eq. (3)). Locally, small perturbations propagate as

$$\delta {{\boldsymbol{\phi }}}\approx J\delta {{\boldsymbol{\theta }}}+{{{\boldsymbol{\varepsilon }}}}_{{{\boldsymbol{\phi }}}},\,J={\nabla }_{{{\boldsymbol{\theta }}}}{{\boldsymbol{\phi }}}\in {{\mathbb{R}}}^{m\times d},$$

(39)

where ε_ϕ accounts for phase noise. The entries of J quantify cross-sensitivities: off-diagonal contributions encode how a perturbation in a given control component (or a perturbation intended to realize one phibit update) can induce phase shifts in other phase channels. However, only a subset of these phase deviations is algorithmically registered at a given step. Let ${{\mathcal{A}}}(\xi )\subset \{1,\ldots ,m\}$ denote the set of phase components that are updated at logical step ξ, and introduce a diagonal selection (projection) matrix

$${P}_{\xi }={{\rm{diag}}}({p}_{1},\ldots ,{p}_{m}),\,{p}_{\ell }=\left\{\begin{array}{ll}1, & \ell \in {{\mathcal{A}}}(\xi ),\\ 0, & \ell \notin {{\mathcal{A}}}(\xi ).\end{array}\right.$$

(40)

Then the registered phase perturbation that can affect the computation at step ξ is

$$\delta {{{\boldsymbol{\phi }}}}_{{{\rm{reg}}}}(\xi )={P}_{\xi }\delta {{\boldsymbol{\phi }}}\approx {P}_{\xi }\left(J\delta {{\boldsymbol{\theta }}}+{{{\boldsymbol{\varepsilon }}}}_{{{\boldsymbol{\phi }}}}\right).$$

(41)

Equation (41) makes the role of the phase cache explicit: phase fluctuations on non-participating phibits are projected out of the logical description at that step, whereas cross-sensitivities among the participating phase channels remain captured.

Under small-noise assumptions with control covariance Σ_θ = Cov(δθ) and readout covariance ${\Sigma }_{{\varepsilon }_{\phi }}={{\rm{Cov}}}({{{\boldsymbol{\varepsilon }}}}_{{{\boldsymbol{\phi }}}})$, the registered phase uncertainty satisfies

$${{\rm{Cov}}}\left(\delta {{{\boldsymbol{\phi }}}}_{{{\rm{reg}}}}(\xi )\right)\approx {P}_{\xi }\left(J{\Sigma }_{\theta }{J}^{{\mathsf{T}}}+{\Sigma }_{{\varepsilon }_{\phi }}\right){P}_{\xi },$$

(42)

so that the effective variance of each updated phase component includes both direct control noise and cross-sensitivity contributions, while non-updated components contribute zero by construction. Robust operation therefore benefits from: reducing control uncertainty and readout noise; improving spectral separation and phase extraction; using redundant phase readouts to average down phase noise.

To connect this model to experimentally observed fluctuations, we estimate the readout-noise term ${\Sigma }_{{\varepsilon }_{\phi }}$ directly from the physical phase cache. Specifically, the step-to-step phase increment errors δϕ = Δϕ − 180^∘ in Supplementary Table 3 provide empirical samples of ε_ϕ for each phibit. The resulting per-phibit standard deviations and within-phibit correlations are summarized in Supplementary Table 1, while the corresponding empirical covariance matrices ${\Sigma }_{{\varepsilon }_{\phi }}^{(i)}$ (in rad²) are reported in Supplementary Table 2. These measurements indicate typical fluctuations at the level of ${{\mathcal{O}}}(1{0}^{\circ }-3{0}^{\circ })$ per cached phase increment, with modest within-phibit cross-correlation.

In the Monte Carlo robustness study, we use these empirical magnitudes to motivate a conservative bounded noise envelope: at each logical stage, every registered phase increment is perturbed independently within  ± 1.35 rad ( ± 77. 4^∘). Within the framework of Eq. (42), this corresponds to treating ${\Sigma }_{{\varepsilon }_{\phi }}$ as being dominated by experimentally observed cache variability (Supplementary Tables 1–2), while allowing additional worst-case scenarios for unmodeled drift. Consequently, the simulated output distributions quantify the tolerance of the full Sylvester-mapped algorithmic sequence to phase noise consistent with the fluctuations measured in the physical cache.

The robustness of the full algorithmic sequence to experimentally observed phase fluctuations is assessed via a Monte Carlo study. As summarized in Fig. 5 and detailed in the Supplementary Note 2, bounded perturbations of the Bloch sphere parameters are applied at each logical update step, and the resulting output distributions are compared to the ideal predictions using the squared statistical overlap. This evaluates how step-to-step phase uncertainty accumulates through the complete multi-stage Sylvester mapping and quantifies the tolerance of the measured probability distributions to phase noise consistent with the experimental fluctuations.