Compressed Sensing for Efficient Fidelity Estimation of GHZ States

IEEE International Conference on Quantum Computing and Engineering (QCE26)

Farrokh Labib¹, David Nicholaeff², Vincent Russo¹, William J. Zeng^1,3
1Unitary Foundation   ²New Mexico Consortium   ³Quantonation

arXiv:2604.27824

vprusso.github.io

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

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 is four matrix elements

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

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: \(C\) is the number to measure.

Fidelity from parity oscillations

The fit returns amplitude \(C \ge 0\) and phase \(\theta\); the fidelity is \(F = (P + C\cos\theta)/2\).

At \(N = 100\): 200 measurement circuits, just to verify the state.

What compressed sensing is

• 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, not 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.

The observation: the signal is sparse

Why random beats regular

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 constant fraction of the wrong candidates: ∼log(candidates) samples leave one survivor.

Recovery separates detection from estimation

• Step 2 is the Lasso¹: Fit all candidate frequencies together, with an \(L_1\) penalty on the cosine and sine coefficients; 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. \(M \propto \log N\) suffices, and the refit leaves only shot noise.

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

The frequency doubles as a certificate

The oscillation frequency is the size of the prepared coherence: a weight-\(w\) coherence oscillates at frequency \(N - 2w\).

• Fixed-frequency fit at \(n = N\): assigns amplitude to a frequency that is not there.
• Blind recovery: returns \(n_{\mathrm{rec}} = N - 2\) and catches the failure.

\(n_{\mathrm{rec}} = N\): the dominant coherence is the one an \(N\)-qubit GHZ state would have.   \(n_{\mathrm{rec}} \neq N\): caught, not mis-certified.
With \(F > 1/2\) this certifies genuine \(N\)-partite entanglement against the phase-optimized target; it does not rule out other errors or smaller competing coherences.

Fixed-frequency fitting produces false certificates

Synthetic corrupted state, emulator sampling: dominant coherence (\(C = 0.8\)) at \(N - 2\), zero coherence at \(N\).

• Known-frequency OLS at \(n = N\) reports median amplitude 0.75 at \(M = 3\), still \(\approx 0.19\) at \(M = 30\): a false-positive entanglement certificate.
• The CS pipeline recovers \(n_{\mathrm{rec}} = N - 2\) and assigns zero coherence to \(N\): the failure is caught.

Recovery saturates below the working budget

• How often does blind recovery return the right frequency? \(P(n_{\mathrm{rec}} = N)\) at \(N = 42\), 200 trials, 1000 shots per angle.
• Two free passes, then a sharp climb: success saturates by \(M \approx 15\).
• Theory supplies the scaling, \(M \propto \log N\); the constant is ours. We budget \(5\ln N \approx 19\) to sit past this knee: a working rule, not a theorem.
• That margin is why the hardware run used 19 angles at \(N = 50\) instead of 100.

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.
• A CNOT copies bit flips, so a fault near the root fans out to its whole subtree.
• Shallow preparation trades depth for high-weight errors.

How one bit flip spreads

What a flag check measures, and what it does not

One check watches half the tree

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 two layers catch different failures

• Flags act before the measurement, on bit flips in the covered region. The certificate 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 certificate is blind to noise that shrinks \(C\) without moving weight, which is what \(F\) is for.
• Flags cost acceptance, not settings: verification is \(M\) circuits either way.

Quantinuum H2-1 hardware, 50 qubits

• 19 random angles instead of \(2N = 100\): 5× fewer circuits; 1000 shots per circuit (cost-limited), hence the wide error bars.
• \(F_{\mathrm{rot}} = (P + C)/2\) scores against the GHZ state with the measured phase folded in; the standard fidelity pays \(\cos\theta\). Rotated fidelity \(\approx 0.72\) with 4 flags; both improve with flags.
• Large phase offset at zero flags: consistent with accumulated coherent calibration error, amplified by \(\theta = \sum_i \epsilon_i\) at \(N = 50\).

Run it yourself: metriq-gym

• 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