IBM Kingston
11.4%95% interval: 9.5–13.6%
Consistent with the ideal probability
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Quantum CFD / Hardware study
From a linear system
to a quantum circuit.
A quantum linear-system solver built from primitives, tested on IBM and Quantinuum hardware, and compared with classical solvers.
Inspect the hardware runs ↓THE COMPUTATIONAL TASKAx = b
Block-encode A
Approximate A⁻¹
Prepare the solution state
Same 2 × 2 solve / Two hardware vendors
What came back from the chips.
Quantinuum Helios-1
7.9%95% interval: 5.8–10.7%
Discrepancy remains undiagnosed
Hardware solution probabilities and confidence intervals
Confidence intervals distinguish shot uncertainty from disagreement with the noiseless prediction. IBM’s interval includes the ideal probability; the Helios-1 interval does not.
| Backend | Shots / selected | P(|1⟩) | 95% interval |
|---|---|---|---|
| Noiseless ideal | — | 11.9% | — |
| IBM Kingston | 1,024 / 949 | 11.4% | 9.5–13.6% |
| Helios-1 | 500 / 469 | 7.9% | 5.8–10.7% |
| Helios emulator | 100 / 98 | 17.3% | 11.1–26.0% |
The numerical toolchain
Check every layer
against linear algebra.
QSVT approximates matrix inversion with a bounded polynomial applied to a block-encoding. This project checks each layer against ordinary linear algebra, runs a small solve on two hardware platforms, and measures the classical baseline on the same matrix families.
Polynomial approximation to the matrix inverse
The polynomial must approximate the reciprocal on the permitted spectral interval while remaining bounded. Conditioning controls how demanding that approximation becomes.
Simulated solver convergence against classical linear algebra
The simulated convergence study tests the solver against classical linear algebra. These checks establish numerical behavior of the toolchain, separate from the hardware measurements.
The engineering constraint
Conditioning sets the cost.
Pressure-correction solves appear repeatedly inside computational fluid dynamics. Their matrices can be sparse yet poorly conditioned, and the polynomial degree grows with the condition number. A modest qubit count therefore does not imply an executable circuit. The measured classical baselines and resource estimates point toward conditioning and usable preconditioners as the research bottleneck.
Hardware observations & CFD resource findings
For the 2×2 hardware solve, IBM Kingston measured 11.4% solution probability against an 11.9% noiseless ideal; its 95% interval includes the ideal.
Quantinuum Helios-1 measured 7.9%, with a 95% interval of 5.8–10.7%. The 3.2σ discrepancy is not explained by shot noise alone.
On the CFD benchmark, even the smallest 16×16 matrix requires polynomial degree 11,357 under the reported cost model. All four classical systems solve in under 1.4 ms.
Block-encoding, polynomial approximation & hardware protocol
Turning inversion into polynomial approximation
QSVT acts on a block-encoding of a matrix and applies a polynomial to its singular values. Approximating the reciprocal implements the core of a linear solve without reading out each eigenvalue. The project checks this chain against ordinary linear algebra before using a small hardware experiment to test the compiled toolchain.
Why the CFD comparison changes the question
Pressure-correction solves appear repeatedly inside computational fluid dynamics. Their matrices can be sparse yet poorly conditioned, and the polynomial degree grows with the condition number. A modest qubit count therefore does not imply an executable circuit. The measured classical baselines and resource estimates point toward conditioning and usable preconditioners as the research bottleneck.
Implementation & checks
- Build block-encoding and polynomial approximation from primitives, checking simulated solutions against NumPy.
- Report hardware probabilities with 95% Wilson confidence intervals, including post-selection counts.
- Reproduce the open Rolls-Royce CFD matrix spectra, estimate polynomial and gate costs, and measure sparse-LU baselines.
Inspect the experiment