Realizing Shor’s algorithm with topological acoustic phase bits

Turning Sound into a New Kind of Computer Bit

Modern quantum computers promise to crack problems, like breaking encryption codes, that overwhelm today’s machines—but they are fragile, expensive, and hard to scale. This work explores a very different path: using carefully controlled sound waves in a solid object to mimic some of the advantages of quantum computing, without needing cryogenic temperatures or exotic hardware. By encoding information in the relative timing, or phase, of these vibrations, the authors show how a table-top acoustic device can carry out the crucial core of Shor’s factoring algorithm, a flagship quantum procedure.

Why Factoring Numbers Matters

Many of the world’s cryptographic systems rely on the fact that breaking a large number into its prime factors is painfully slow on ordinary computers. Shor’s algorithm, proposed in the 1990s, shook this foundation by showing that a quantum computer could, in principle, factor such numbers dramatically faster. At the heart of the algorithm lies a task called period finding: detecting a hidden repeating pattern in a mathematical function. Quantum machines tackle this by preparing many possible inputs at once, letting them interfere, and then reading out a pattern that reveals the period. Recreating this trick in a more accessible, classical system could offer a new way to explore such powerful algorithms without needing a full-scale quantum computer.

From Vibrating Rods to Phase Bits

The team builds their “computer” from three aluminum rods glued together to form an acoustic metastructure. When these rods are driven by ultrasonic transducers, they support many vibrational modes that can interact in nonlinear ways. Mixing two driving tones produces a host of new frequencies, each acting as a distinct channel. Instead of representing bits as voltages or quantum spins, the authors define phase bits, or “phibits,” where the logical state lives in the relative phase of vibrations between rods at one of these mixing frequencies. Because phase differences can be measured directly, each phibit connects an abstract mathematical state to something that can be observed and adjusted in the lab.

Keeping Track of Many Moving Phases

A single change to the drive frequency nudges the phase of every vibrational channel at once, which is useful for coupling them but dangerous for precise logic. To tame this, the authors introduce a “phase cache,” a bookkeeping scheme that records which phibits are supposed to change at each computational step. The cache distinguishes between the raw physical phases that are always shifting and the logical phases that count as operations in the algorithm. It can even associate different frequency ranges with different logical steps for each phibit, effectively carving a smooth, continuous frequency sweep into a sequence of well-defined gates. This makes it possible to scale up the number of phibits while keeping the logical description manageable.

Bridging Physical Actions and Logical Gates

To use phibits for serious computation, the authors must ensure that a given frequency manipulation truly behaves like a desired logic gate, such as a rotation or a controlled operation. They frame this as a mathematical matching problem: for each step in the circuit, a matrix describing the target gate must line up with a matrix describing the actual phase shifts produced by the device. They solve this using a technique called operator spectra shift, which slightly adjusts the description of the physical operation so that a unique mapping exists between physical changes and logical gates. Chaining these mappings across many steps yields a complete translation from drive-frequency trajectories to the abstract circuit that implements the period-finding routine.

Factoring Numbers with Sound

Armed with phase bits, the phase cache, and the gate-mapping framework, the researchers implement the period-finding core of Shor’s algorithm on their acoustic platform. They factor the number 15 using several different bases, and, more strikingly, factor 35 for a choice of parameters that has proven challenging for quantum hardware demonstrations. Instead of repeatedly measuring a delicate quantum state, they reconstruct the final outcome probabilities from a single run by reading out all relevant phases and feeding them through their mathematical mapping. Monte Carlo studies, which inject sizeable random phase errors at every step, show that the resulting probability distributions stay very close to the ideal ones, indicating strong robustness to realistic noise in the acoustic device.

What This Means for Future Computing

For non-specialists, the main message is that you do not always need a full-blown quantum computer to harness some quantum-style advantages. By cleverly using sound waves and their phases, this work realizes the key engine inside Shor’s factoring algorithm on a room-temperature, classical system built from standard components. The computation relies not on fragile quantum entanglement, but on strong classical correlations between many vibrational modes and on careful bookkeeping of how their phases evolve. While this approach will not replace true quantum computers for every task, it opens a promising route to explore powerful algorithms and specialized computing devices that sit between conventional electronics and full quantum hardware.

Citation: Kuk, I., Djordjevic, I.B., Runge, K. et al. Realizing Shor’s algorithm with topological acoustic phase bits. Commun Eng 5, 60 (2026). https://doi.org/10.1038/s44172-026-00623-6

Keywords: quantum-inspired computing, topological acoustics, Shor's algorithm, phase-based information, nonlinear metastructures