Experimental Benchmarking of an Automated Deterministic Error-Suppression Workflow for Quantum Algorithms

Excitement about the promise of quantum computers is tempered by the reality that the hardware remains exceptionally fragile and error prone, forming a bottleneck in the development of alternative applications. In this paper, we describe and experimentally test a fully autonomous workflow designed to deterministically suppress errors in quantum algorithms from the gate level through to circuit execution and measurement. We introduce the key elements of this workflow, delivered as a software package called Fire Opal, and survey the underlying physical concepts: error-aware compilation, automated system-wide gate optimization, automated dynamical decoupling embedding for circuit-level error cancellation, and calibration-efficient measurement-error mitigation. We then present a comprehensive suite of performance benchmarks executed on IBM hardware, demonstrating up to $>1000\times$ improvement over the best alternative expert-configured techniques available in the open literature. Benchmarking includes experiments using up to 16 qubit systems executing the following: Bernstein Vazirani, quantum Fourier transform, Grover’s search, quantum approximate optimization algorithm, variational quantum eigensolver, syndrome extraction on a five-qubit quantum error-correction code, and quantum volume. Experiments reveal a strong contribution of Non-Markovian errors to baseline algorithmic performance; in all cases the deterministic error-suppression workflow delivers the highest performance and approaches incoherent error bounds without the need for any additional sampling or randomization overhead, while maintaining compatibility with all additional probabilistic error suppression techniques.

Figure 1

The Fire Opal automated error-suppressing workflow for quantum algorithms. Each step (described in the main text) is executed autonomously with no user configuration, given an arbitrary input circuit and access to a supported quantum hardware backend. Measurement error mitigation is performed after the hardware backend returns raw results, and the output is returned to the user.

Figure 2

Success probability of a BV algorithm as a function of the number of qubits. Default, expert, and Q-CTRL settings are described in Table 2. Associating the mode of the distribution with the correct answer, the Q-CTRL pipeline returns the correct answer for all system sizes while the other methods fail to return the correct answer beyond eight qubits (default) and beyond ten qubits (expert).

Figure 3

Success probability of the BV algorithm for the different settings as appeared in Fig. 2, together with two projected performance bounds, which assume the existence of either uncorrelated gate errors or $T_1$ processes only. The gate-error limit is calculated by a simple multiplication of the fidelities derived from the backend for all single and two-qubit gates in the circuits. The circuit $T_1$ limit is calculated by a multiplication of the $T_1$-limited fidelity of each participating qubit, calculated as $\prod _i\exp (-T_i/2T_1^{(i)})$; here $T_i$ is the total active time of qubit $i$ (from first operation until measurement).

Figure 4

The mean success probability, over ten different initial states, of QFT circuits as a function of the system size. Absolute performance is lower than for BV due to the deeper circuit structure.

Figure 5

(a)–(c) The output probability distributions over all possible 5Q bitstrings after two iterations of a Grover search algorithm with a specific oracle (00101 as a solution). (a) The ideal expected distribution as calculated using a noiseless simulator, and (b),(c) are probabilities obtained using default settings (black) and using the Q-CTRL pipeline (purple). The resemblance between the measured and ideal distributions is captured by the circuit infidelity. The distribution obtained by using Q-CTRL tools is $8\times$ closer to the ideal results. Similar improvement is observed for all target states. (d) Selectivity calculated for all target bitstrings, demonstrating $S>1$ for all possible five qubit states using the Q-CTRL pipeline compared to $S<0$ using the default settings.

Figure 6

The cost (expectation of $H$) landscape of a $p=3$ QAOA circuit representing a seven-node and ten-edge weighted MaxCut problem. Here, we compare the ideal landscape (top), the experimentally obtained landscapes using the Q-CTRL pipeline (middle) and using the default settings (bottom). The ideal optimum is denoted by the red dot. Structural similarity relative to the ideal landscape is calculated and shown for each setting (see text).

Figure 7

VQE performance with different workflows. (a) Computed ground-state energy of the ${\mathrm{Be}\mathrm{H}}_2$ molecule as a function of the Be—H bond distance for a six-qubit ansatz. The dotted line displays results from an ideal simulator, and the individual points represent the results computed on real hardware with different settings. (b) Deviation from expected Pauli measurement as a function of the ideal value. Perfect agreement between theory and experiment would correspond to all points on the zero dashed line. To quantify the agreement, we calculate the Pearson distance, $P_d$ for the three methods: default, 0.177; expert, 0.042; Q-CTRL, 0.011.

Figure 8

Performance of QEC using the Q-CTRL pipeline. (a)–(d) Benchmarking data for the five-qubit repetition code. (a),(b) Joint probability distribution of the measured and expected syndromes for the different settings after an error was injected to qubit 2. The syndrome bit values “0” or “1” correspond to the measurement outcomes “$+1$” and “$-1$” for each of the stabilizer generators in the ordered set, $G$, in Eq. (1). (c) Detection success as a function of the injected error for the default and Q-CTRL pipelines using color-coding consistent with previous graphs. Dashed line denotes the randomness threshold, which represents the success achieved by a random chance. (d) Mean detection success averaged over the six bit-error-location possibilities, comparing the two protocols. Arrow indicates quantitative performance enhancement, calculated with respect to the dashed line. (e),(f) Data on full five-qubit QEC. (e) Detection success for each of the four stabilizers measured for the different pipelines. As before, the dashed line is the random chance value. (f) Overall mean (over injected errors) detection success incorporating data averaged over all four stabilizers simultaneously with net performance enhancement with respect to the random value represented.

Figure 9

A quantum volume experiment on a 16-qubit IBM device with reported QV of 32. In the experiment, we generate 300 random circuit instances and sample each 1000 times. We then calculate the mean heavy output probability over the 300 random circuits. (a) The mean HO probability versus the number of qubits as obtained using both expert and Q-CTRL settings. (b) The cumulative sum and standard error as extracted from bootstrapping the 300 data points obtained in the 6Q experiment.

Figure 10

Success probability (on a logarithmic scale) of the BV algorithm on three different dates (all different than the plot in the main text). Data presented on a log scale.

Figure 11

We compare the different compilation methods using the 16-qubit QFT circuit. SABRE is a stochastic method, therefore, we present $50$ instances of the method. We use here the total number of CX gates and the total circuit duration as comparison features. As seen, the Q-CTRL method typically finds shorter circuits with a lower number of entangling gates.

Figure 12

A Ramsey-like experiment where three coupled qubits are prepared at the $|+⟩$ state (i.e., $⟨X⟩=1$). The state should remain unchanged under zero error conditions. Optimized DD significantly reduces the idling error when compared to the standard DD protocol.

Figure 13

An example of the DD embedding. The input circuit, on the top, is a compiled circuit and the output circuit of our automated scheme, on the bottom, is the original circuit with the optimal DD structure embedded in it.

Figure 14

(Figure adapted from Ref. [6].) A comparison between a default cnot gate and an optimized gate found by a simulated annealing optimizer. Each gate is applied $N$ times on two different initial states and the infidelity with respect to the ideal state was calculated by performing a full-state tomography. The gate errors are extracted from the slope of the infidelity, showing an approximately $2.3\times$ improvement compared to the default.

Figure 15

Full device calibration can be done in parallel requiring a fixed number of steps depending on the device connectivity. Heavy-hex topology can be tuned up in only four steps irrespective of the device size.

Figure 16

Comparison of readout error, denoted by the Hellinger loss, averaged over 2000 random probability distributions on a seven-qubit quantum computer. The complete method includes a full calculation of the confusion matrix, which requires $2^7$ experiments to construct the filter.