🔬 applied physics

A 2048-spin bulk acoustic wave Ising machine for number partitioning and Sudoku

This paper presents a compact, low-power, and thermally stable 2,048-spin bulk acoustic wave Ising machine that utilizes solid-state delay lines to achieve all-to-all connectivity and efficiently solve complex optimization problems like MAX-CUT, number partitioning, and Sudoku, outperforming both state-of-the-art optical machines and simulated algorithms in stability and specific problem-solving capabilities.

Original authors: Venkatesh Vadde, Roman Ovcharov, Victor H. González, Roman Khymyn, Artem Litvinenko, Johan Åkerman

Published 2026-07-03

📖 5 min read🧠 Deep dive

CC BY 4.0

Original authors: Venkatesh Vadde, Roman Ovcharov, Victor H. González, Roman Khymyn, Artem Litvinenko, Johan Åkerman

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to solve a massive, incredibly difficult puzzle. You have thousands of pieces, and you need to arrange them so that they fit together perfectly to minimize "friction" or "tension." In the world of computers, these are called optimization problems (like figuring out the best way to split a group of people into two teams so the teams are as balanced as possible, or solving a Sudoku).

For a long time, regular computers have struggled with these puzzles because the number of possible arrangements grows so fast that it would take them longer than the age of the universe to find the best answer.

This paper introduces a new kind of "puzzle solver" called a Bulk Acoustic Wave Ising Machine (BAWIM). Think of it as a specialized, physical machine built to solve these puzzles by using sound waves instead of just electricity and code.

Here is a simple breakdown of how it works and why it's special:

1. The Core Idea: A Ring of Sound

Imagine a circular racetrack. Instead of cars, we send tiny packets of sound waves (specifically, sound traveling through a solid block of quartz crystal) around this track.

The Spins: Each "lap" or pulse of sound represents a single piece of the puzzle, called a "spin." It can be in one of two states: like a coin showing Heads (+1) or Tails (-1).
The Track: The machine uses a delay line (a tube for sound) that is about 700 microseconds long. By sending pulses one after another, the machine can hold 2,048 of these "spins" at the same time.
The Loop: These sound pulses circulate in a loop. The machine constantly listens to the pulses, calculates how they should interact with each other based on the rules of the puzzle, and then "injects" a little push to change their state if needed.

2. The "Coach" (The Feedback System)

The machine has a smart coach (an FPGA computer chip) watching the race.

The coach measures the phase (timing) of the sound waves.
It calculates the "rules" of the specific puzzle (e.g., "Spin A and Spin B should be opposite").
It sends a signal back into the loop to nudge the sound waves toward the correct arrangement.
Over time, the sound waves naturally settle into the lowest-energy state, which corresponds to the best solution for the puzzle.

3. Why This Machine is a Game-Changer

The authors compare their new machine to the current "champions" of this field, which are Optical Coherent Ising Machines (CIMs). These older machines use light (lasers) and fiber optics. Here is how the new sound-based machine wins:

The "Thermostat" Problem:
- The Old Way (Light): Imagine trying to keep a giant, 5-kilometer-long fiber-optic cable perfectly stable. If the room temperature changes by just one degree, the light gets out of sync, and the whole machine breaks. You need a massive, expensive, water-filled temperature-controlled box to keep it working.
- The New Way (Sound): The BAWIM is like a tiny, solid quartz crystal. It is incredibly stable. The paper claims it is 10,000 times more stable against temperature changes than the light-based machines. You can put it on a regular desk in a normal room, and it works perfectly without needing a giant thermostat.
Size and Cost:
- The light machines are huge, expensive, and consume a lot of power.
- The BAWIM is a "tabletop" device. It fits on a desk, uses standard off-the-shelf electronic parts (like those found in your phone or radio), and uses very little power (about 9.5 watts, roughly the same as a bright lightbulb).

4. What Puzzles Did They Solve?

The researchers tested their machine on three types of challenges:

MAX-CUT: A classic math problem about splitting a network into two groups. The machine found a very good solution in about 341 milliseconds (less than half a second).
Number Partitioning: Dividing a list of random numbers into two groups with equal sums. The machine solved this better and faster than a top-tier software algorithm called "Simulated Bifurcation."
Sudoku: They even solved a Sudoku puzzle. This is tricky because Sudoku requires exact answers (no "almost right" allowed). The machine successfully found the correct solution, whereas the software algorithm failed to find a valid one.

5. The Future Potential

The paper suggests that if they swap their current 20 MHz sound frequency for a much faster 16 GHz one (which is possible with existing technology), the machine could solve these problems in less than a millisecond.

Summary

In short, the authors built a tiny, stable, low-power, desk-top computer that uses sound waves to solve complex math puzzles. Unlike the current giant, temperature-sensitive laser machines, this new device is robust, cheap to build, and works reliably in a normal room. It proves that we can solve hard optimization problems using simple, solid-state physics rather than massive, fragile optical setups.

Technical Summary: A 2048-Spin Bulk Acoustic Wave Ising Machine

Problem Statement
The advancement of modern computing is increasingly constrained by the limitations of traditional silicon-based architectures, particularly when addressing non-deterministic polynomial-time hard (NP-hard) and NP-complete problems. Combinatorial optimization tasks, such as MAX-CUT, number partitioning, and Sudoku, require solution times that scale exponentially with problem size. While various analog Ising machines have been developed using platforms like superconducting qubits, nanomechanical systems, and photonic coherent Ising machines (CIMs), significant barriers remain. Specifically, optical CIMs, despite demonstrating high spin scalability (up to 100,000 spins), suffer from large physical footprints, high power consumption, poor thermal stability, and high costs, often confining them to laboratory environments. Furthermore, previous acoustic wave-based approaches (e.g., Surface Acoustic Wave Ising Machines) have been limited to small spin counts (e.g., 50 spins) and low coupling resolution (1-bit).

Methodology
The authors present a time-multiplexed Ising machine based on Bulk Acoustic Waves (BAWs) propagating through solid-state delay lines. The core architecture utilizes two serially connected quartz-based BAW delay lines, each with a 707 µs delay and a 3-dB bandwidth centered at 20.5 MHz. This configuration supports a total of 2,124 RF pulses, of which 2,048 are utilized as Ising spins.

Key design features include:

Time-Multiplexing: Spin states (+1 and -1) are encoded in the relative phase of propagating RF pulses. A pulse period of 665 ns (corresponding to 6–7 cycles of the carrier) is enforced via a high-speed RF switch.
Phase Binarization: A Phase-Sensitive Amplifier (PSA) acts as a time-gate, enforcing bistable phases (0° or 180°) relative to the pumping signal, effectively binarizing the analog wave packets.
All-to-All Connectivity: A Measurement-and-Feedback Block (MFB) extracts a portion of the circulating pulses, measures their phase using a phase detector, and processes the data via an FPGA. The FPGA computes the coupling terms ( $c_i = \sum J_{ij}s_j$ ) with 15-bit resolution and reinjects the weighted coupling signals back into the loop.
Stability Mechanism: The system satisfies the Barkhausen stability criteria through linear amplifiers compensating for the 36 dB attenuation of the delay line and a reference signal ensuring integer multiples of $2\pi$ phase shift. The BAW medium offers superior thermal stability compared to optical fibers.

Key Contributions

Scalability and Resolution: The machine successfully hosts 2,048 spins with 15-bit coupling resolution, a significant improvement over previous acoustic wave implementations which were limited to ~50 spins and 1-bit resolution.
Thermal Stability: The design achieves a temperature coefficient of phase accumulation of 780 deg/°C, which is approximately four orders of magnitude lower than state-of-the-art optical CIMs (1.73 × 10⁷ deg/°C). This eliminates the need for complex, active temperature control systems, allowing operation on a standard tabletop at room temperature.
Hardware Efficiency: The system operates with a high spin duty cycle (50%) and requires only 6 carrier periods per spin, contrasting with optical CIMs that may require thousands of periods and lower duty cycles.
Problem Versatility: The machine is demonstrated to solve not only standard graph problems (MAX-CUT) but also complex, constraint-satisfaction problems like Number Partitioning and Sudoku, which require exact solutions rather than approximate ground states.

Results

MAX-CUT Performance: On a 2,048-node graph with 10% edge density, the BAWIM reached 90% of the best Ising energy found by the Heated-Ballistic Simulated Bifurcation (HbSB) algorithm in 341 ms. While HbSB reached the same energy level in 102 ms on a high-end CPU, the BAWIM demonstrated robust performance across varying edge densities (25% to 100%), achieving up to 99.99% of the optimal MAX-CUT score.
Number Partitioning: For problem sizes ranging from 32 to 2,048 integers, the BAWIM achieved a 100% success rate for 99.9% approximate solutions in most instances, outperforming the HbSB algorithm.
Sudoku: The machine successfully solved a 9×9 Sudoku puzzle, reaching the ground state and producing a valid solution. In contrast, the HbSB algorithm failed to produce correct solutions for the same instance.
Power Consumption: The total power consumption is approximately 9.57 W, dominated by the FPGA and amplifiers. The authors note that replacing standard amplifiers with specialized low-power components could reduce consumption to roughly 5.9 W.
Scalability Potential: The authors project that utilizing higher frequency delay lines (e.g., 16.455 GHz) could reduce the solution time to approximately 0.46 ms, representing a ~750-fold speedup over the current 20.5 MHz implementation.

Significance and Claims
The paper claims that the Bulk Acoustic Wave Ising Machine (BAWIM) represents a significant step toward practical, commercially viable combinatorial solvers. By leveraging solid-state delay lines, the system overcomes the thermal instability and environmental constraints that plague optical CIMs, offering a robust, low-power, and tabletop solution. The authors assert that their design provides four orders of magnitude higher thermal stability than existing coherent Ising machines and demonstrates superior performance over the simulated bifurcation algorithm on complex, non-native constraint problems like Sudoku. The work highlights the potential for BAWIMs to serve as a reliable, energy-efficient architecture for solving NP-hard problems without the need for cryogenic cooling or complex stabilization infrastructure.