Compressed sensing for efficient fidelity estimation of GHZ states

IEEE International Conference on Quantum Computing and Engineering (QCE26)

Farrokh Labib¹, David Nicholaeff², Vincent Russo^1,4, William J. Zeng^1,3
1Unitary Foundation   ²New Mexico Consortium   ³Quantonation   ⁴IonQ

arXiv:2604.27824

vprusso.github.io

paper

GHZ states as a hardware benchmark

\[ |\mathrm{GHZ}\rangle = \frac{1}{\sqrt{2}} \bigl(|0\rangle^{\otimes N} + |1\rangle^{\otimes N}\bigr) \]

secret sharing metrology error correction

Fidelity \(F\): how close the built state is to the intended one, from 0 to 1.

This talk: verify GHZ states with \(\log N\) measurement circuits instead of \(2N\),
and detect errors during preparation on trapped-ion hardware.

¹Javadi-Abhari, Ali, et al. “Big cats: entanglement in 120 qubits and beyond.” arXiv:2510.09520 (2025).

Fidelity as population and coherence

\[ F(\rho) = \langle \mathrm{GHZ} | \rho | \mathrm{GHZ} \rangle = \frac{1}{2}\Bigl( \underbrace{\langle\mathbf{0}|\rho|\mathbf{0}\rangle + \langle\mathbf{1}|\rho|\mathbf{1}\rangle}_{ \text{population } P} + \underbrace{\langle\mathbf{0}|\rho|\mathbf{1}\rangle + \langle\mathbf{1}|\rho|\mathbf{0}\rangle}_{ \text{coherence}} \Bigr) \]

Here \(\mathbf{0} = 0\cdots0\) and \(\mathbf{1} = 1\cdots1\). Diagonal entries are outcome probabilities; the corner off-diagonals measure the superposition itself.

A coin flip scores exactly \(F = 1/2\). Everything above \(1/2\) lives in the orange corners: the coherence is the number to measure.

Measuring population and coherence

Average parity follows \(\mathcal{P}(\phi) = C\cos(N\phi + \theta)\); the amplitude \(C \ge 0\) and phase \(\theta\) give \(F(\rho) = (P + C\cos\theta)/2\).

\(Z\) counts cannot see the off-diagonals; after the rotation the coherence shows up as an even/odd bias in the outcomes.

Resolving any frequency blind takes \(2N\) angles^1,2: 201 circuits in all at \(N = 100\).

¹Nyquist, “Certain topics in telegraph transmission theory,” Trans. AIEE 47.2 (1928).
²Shannon, “Communication in the presence of noise,” Proc. IRE 37.1 (1949).

The parity signal is sparse

Compressed slide on compressed sensing

• The right-hand chart is the signal's recipe: the signal rewritten as a sum of simple waves, one entry per wave saying how much of it the signal contains. Natural signals have nearly empty recipes.
• A JPEG keeps the few big entries and looks identical, but it measures everything first: fine when a measurement is a cheap pixel.
• Compressed sensing pays for the information rather than the signal length: few random samples, solve for the big entries.¹ Ours is the expensive case: each measurement is a quantum circuit.

¹Brunton, Steven L., and J. Nathan Kutz. Data-Driven Science and Engineering. Cambridge University Press, 2019.

Random sampling avoids aliasing

Frequency 2, sampled at 8 equally spaced angles.

• Frequency 10 passes through every one of the same samples: \(\cos(10\phi_k) = \cos(2\phi_k)\) exactly. A regular grid has a period for impostors to hide behind; ruling that out for every frequency up to \(N\) is what costs \(2N\).
• A random angle has no period: distinct frequencies agree there with probability zero, so each further sample rules out a roughly constant fraction of the wrong candidates: ∼log(candidates) samples leave one survivor.

Recovery pipeline via compressed sensing

• Lasso for which frequencies, least squares for how big: the penalty that isolates the frequency also shrinks its amplitude, so the refit is unpenalized.
• The Lasso¹ fits all candidate frequencies together, with an \(L_1\) penalty; weak candidates are driven exactly to zero.
• Why so few: amplitude and phase absorb the first samples, then each further one rules out a constant fraction of the wrong frequencies, so \(M \propto \log N\) suffices.

¹Tibshirani, Robert. “Regression shrinkage and selection via the lasso.” Journal of the Royal Statistical Society B 58.1 (1996).

Choosing the measurement budget

• Recovery improves rapidly with \(M\), then saturates around \(M \approx 15\) for \(N = 42\).
• Choose enough random angles to reach the recovery plateau; beyond that, additional angles give diminishing returns.
• The required budget scales as \(M = O(\log N)\); our rule \(M \approx 5\ln N\) gives 19 angles at \(N = 50\).

Preparing a GHZ state: chain versus tree

• Same state, same gate count: only the CNOT schedule differs; every layer is an opportunity for noise, so shallower wins.
• Shallow preparation trades depth for high-weight errors.

How one bit flip spreads

Flag checks

Coverage of a single check

¹Martiel, Simon, and Ali Javadi-Abhari. “Low-overhead error detection with spacetime codes.” arXiv:2504.15725 (2025).

²Javadi-Abhari, Ali, et al. “Big cats: entanglement in 120 qubits and beyond.” arXiv:2510.09520 (2025).

The complete protocol

• Flags reduce the errors we keep; compressed sensing reduces the measurements we use.
• Flags act before the measurement, on bit flips in the covered region. The frequency check acts after, on whatever survived, by checking the size of the coherence that is left.
• Neither is complete: flags are blind to phase errors and to flips that reach both checked qubits; the frequency check is blind to noise that shrinks \(C\) without moving weight, which is what \(F\) is for.
• Flags cost acceptance rather than settings: verification is \(M\) circuits either way.

Quantinuum H2-1 hardware, 50 qubits

• \(19\) random angles instead of the blind scan's \(2N = 100\): \(5\times\) fewer circuits; \(1000\) shots per circuit (cost-limited), hence the wide error bars.
• Fidelity rises as flags are added: \(0.667 \to 0.708\) with \(4\) flags.
• More flags cover more of the tree, but each adds gates and costs acceptance, so we stopped at \(4\).

Run it yourself: metriq-gym

{
  "benchmark_name": "GHZ",
  "num_qubits": 26,
  "method":
    "compressed_sensing",
  "num_phases": 20,
  "shots": 30000
}

• One dispatch, one poll, on any supported backend (IBM, Quantinuum, IonQ, local simulators); the estimator and the \(n_{\mathrm{rec}} = N\) size check come built in.
• DFE and full parity oscillation are included as alternative verification methods for comparison.

github.com/unitaryfoundation/metriq-gym