Verifiable measurement-based quantum random sampling with trapped ions

TL;DR

This work tackles verifiable quantum random sampling on near-term devices by leveraging measurement-based quantum computing (MBQC) with large, randomly rotated cluster states generated in trapped-ion processors. It introduces efficient fidelity certificates via direct fidelity estimation (DFE) generalized to MBQC, and employs qubit recycling to sample from cluster states larger than the physical register, enabling scalable verification up to practical sizes. The authors connect average-case MBQC fidelity to cross-entropy benchmarking (XEB) while establishing conditions under which XEB yields reliable fidelity estimates, and provide a Stockmeyer-based threshold arguing for hardness of classical verification beyond a fidelity threshold. Overall, the study demonstrates a feasible path toward verified quantum advantage in random sampling using MBQC, with quantitative benchmarking, noise-robust certificates, and concrete experimental progress toward scalable, verifiable quantum computation.

Abstract

Quantum computers are now on the brink of outperforming their classical counterparts. One way to demonstrate the advantage of quantum computation is through quantum random sampling performed on quantum computing devices. However, existing tools for verifying that a quantum device indeed performed the classically intractable sampling task are either impractical or not scalable to the quantum advantage regime. The verification problem thus remains an outstanding challenge. Here, we experimentally demonstrate efficiently verifiable quantum random sampling in the measurement-based model of quantum computation on a trapped-ion quantum processor. We create and sample from random cluster states, which are at the heart of measurement-based computing, up to a size of 4 x 4 qubits. By exploiting the structure of these states, we are able to recycle qubits during the computation to sample from entangled cluster states that are larger than the qubit register. We then efficiently estimate the fidelity to verify the prepared states -- in single instances and on average -- and compare our results to cross-entropy benchmarking. Finally, we study the effect of experimental noise on the certificates. Our results and techniques provide a feasible path toward a verified demonstration of a quantum advantage.

Figures (12)

• Figure 1: Overview of the experiment.(a) Sketch of the ion trap quantum processor. Strings of up to 16 ions are trapped in a linear chain. Any single ion or pair of ions can be individually addressed by means of steerable, tightly focused laser beams (dark red) to apply resonant operations $\mathrm{R}_j$ or Mølmer-Sørensen entangling gates $\mathrm{MS}_{i,j}$. Global detection, cooling (blue), and repumping (pink) beams are used to perform a mid-circuit reset of part of the qubit register Ringbauer2021. (b) Implemented cluster states. Cluster states with local rotation angles $\beta_{i} \in \{0, \frac{\pi}{4}, \ldots, \frac{7\pi}{4}\}$ up to a size of $4 \times 4$ qubits are created in the qubit register. Each cluster state is defined by its $N$ stabilizers $S_k$ which are given by rotated $X$ operators $\tilde{X}_k = X_k(\beta_k)$ at each site $k = 1, \ldots, N$ multiplied with $Z$ operators on the respective neighbouring sites. (c) Recycling of qubits. Using sub-register reset of qubits, we prepare cluster states that are larger than the qubit register. For example, using four ions, we prepare cluster states of size $2 \times 3$. (d) Single-instance verification. In order to verify single cluster state preparations with fixed rotation angles $\beta$, we measure it in different bases. To perform fidelity estimation we measure uniformly random elements of its stabilizer group, which is obtained by drawing a random product of the $N$ stabilizers $S_k$, indexed by a length-$N$ random bitstring indicating for each $S_k$ whether it participates in the product. To sample from the output distribution, we measure in the $X$-basis. These samples are verified in small instances by the empirical total-variation distance (TVD). (e) Average-case verification. To assess the average quality of the cluster state preparations, we perform measurements on cluster states with random rotations. By measuring a random element of the stabilizer group of each random cluster state, we obtain an estimate of the average fidelity. From the samples from random cluster states in the $X$-basis, we compute the cross-entropy benchmark (XEB) by averaging the ideal probabilities $p_\beta(x)$ corresponding to the samples $x$ and the cluster with angles $\beta$.
• Figure 2: Sketch of the circuit with qubit recycling.(a) Recycling. After detection, a measured qubit is either still in the $\vert{1}\rangle$ state ("dark" outcome) or in one of the two S levels ("bright" outcome). We reset it to the $\vert{0}\rangle$ state by first applying an addressed $\pi$-pulse (1) on the $\vert{S'}\rangle\rightarrow\vert{D'}\rangle$-transition. A subsequent global 854 quench pulse (2) transfers population from all D-levels to the P$_{3/2}$ manifold, from which (3) spontaneous decay occurs, preferentially to the $\vert{0}\rangle$ state in the S manifold. We repeat this process twice, which is sufficient to return about 99% of the population to the $\vert{0}\rangle$ state. (b) Circuit. The individual qubits are prepared in a product state depending on the random angles $\beta_i$ and entangled via $XX$ interactions and some single-qubit gates (white boxes) to create a cluster state; see the SI, \ref{['app:compiled circuits']} for details. The measurement of the qubits is achieved by exciting the $P \leftrightarrow S$ transition. In order to perform a circuit with recycling, a coherent $\pi$-pulse on the $S \rightarrow D'$ transition (denoted by H$_S$) is applied to 'hide' the qubits which should not be measured in the D-manifold. After the measurement, the chain is cooled using polarization-gradient cooling. The reset makes use of local pulses on the measured qubit that transfer the remaining population of $\vert{S'}\rangle$ to the D$_{5/2}$-manifold (denoted by P) and global pulses that transfer the population of that manifold back to $\vert{0}\rangle$. Prior to the reset, all unmeasured qubits are 'hidden' in the S$_{1/2}$-manifold. For this, the population which was in $\vert{0}\rangle$ prior to the measurement is coherently transferred back to $\vert{S}\rangle$ via a $\pi$-pulse (H$_S^{-1}$), and the population which is in $\vert{1}\rangle$ is transferred to $\vert{S'}\rangle$ via a $\pi$-pulse on the $D\rightarrow S'$ transition (H$_D$). After the reset procedure (a), a $\pi$-pulse (H$_D^{-1}$) is applied to the unmeasured qubits to transfer the population which was previously in $\vert{1}\rangle$ back from $S'$.
• Figure 3: Experimental results for single-instance verification. Root infidelity estimate $\sqrt{1- F}$ (hexagons), and empirical TVD (stars) for single instances of random MBQC cluster states with recycling (blue) and without (pink). Note that the horizontal axis is labelled with the cluster size $n\times m$ and scaled with qubit number $n\, m$. The root infidelity upper-bounds the TVD per \ref{['eq:estimate tvd relation']}. Colored error bars represent the 3$\sigma$ interval of the statistical error. Uncorrelated measurement noise reduces or increases the measured state fidelity compared to the true fidelity asymmetrically depending on its value, such that the shown values are lower bounds to the true state fidelity, see the Methods section for details. The worst-case behaviour of the measurement noise is represented by gray error bars. In the non-recycling experiment, the register size is increased between the $2\times 3$ and the $3\times 3$ instance, leading to a decrease in the local gate fidelities. Modeling the noise as local depolarizing noise after each entangling gate (dotted lines), we obtain effective local Pauli error probabilities after the two-qubit gates of 5.3%, 2.6%, and 1.0%, for the recycling data, for the large-register non-recycling data, and the small register non-recycling data, respectively; see \ref{['app:average fidelity']} of the SI. The shaded green area is the acceptance region corresponding to an infidelity threshold of 8.6% arising in the rigorous hardness argument as sketched in \ref{['app:stockmeyer argument']} of the SI. Since the accuracy of the TVD estimate scales with the system dimension already for cluster sizes of $4\times3$ and $4\times4$ infeasible amount of samples would be required for an accurate estimate, and hence these are not shown. See \ref{['tab:num samples']} of the SI for experimental details.
• Figure 4: Single-instance verification with artificially added phase noise. Root infidelity estimate $\sqrt{1-F}$ (hexagons) and empirical total-variation distance $d_{\text{TV}}$ (stars) of a $2\times 2$ cluster state with artificially introduced local (pink) and global (green) phase noise---$Z$-rotations with rotation angle drawn from a Gaussian distribution with variance $\sigma^2$---before and after Mølmer-Sørensen gate applications as a function of the noise strength $\sigma$, see Methods for details. Solid (dashed) lines show simulated root infidelity (total-variation distance) for the respective types of noise. The experimental data (top axis) is shifted with respect to the simulations (bottom axis) due to the fact that there is residual noise when no artificial noise is introduced. The value of the relative shift given by $0.045\pi$ (dashed vertical line) provides an estimate for the natural noise strength. Colored error bars represent the 3$\sigma$ interval of the statistical error. The systematic measurement error of the fidelity estimate is represented by gray error bars.
• Figure 5: Experimental results for average performance verification. Average fidelity estimate from direct fidelity estimation (DFE) (pink hexagons), from linear XEB (triangles), and from logarithmic (log) XEB (diamonds, see Methods for the definition) using $1000$ random cluster states and $50$ shots per state. Based on calibration data for the gate fidelities of single-qubit gates $f_{1Q} = 99.8\%$, two-qubit gates $f_{2Q} = 97.5 \pm 0.5\%$, and measurements $f_{M} = 99.85\%$, we compute a prediction for the fidelity (gray shaded line). We extract an effective local Pauli error probability of $1.7\%$ (dotted line), see \ref{['app:average fidelity']} of the SI. Colored error bars represent the statistical $3\sigma$ error. For uncorrelated measurement noise, the fidelity estimate provides a lower bound to the true state fidelity. Gray error bars represent the worst-case systematic measurement error.