Error-mitigated data-driven circuit learning on noisy quantum hardware

Abstract

Application-level benchmarks measure how well a quantum device performs meaningful calculations. In the case of parameterized circuit training, the computational task is the preparation of a target quantum state via optimization over a loss landscape. This is complicated by various sources of noise, fixed hardware connectivity, and generative modeling, the choice of target distribution. Gradient-based training has become a useful benchmarking task for noisy intermediate-scale quantum computers because of the additional requirement that the optimization step uses the quantum device to estimate the loss function gradient. In this work, we use gradient-based data-driven circuit learning to qualitatively evaluate the performance of several superconducting platform devices and present results that show how error mitigation can improve the training of quantum circuit Born machines with 28 tunable parameters.

Appendices

Appendix A: Mitigation SWAP gate noise

The motivation behind the design of circuit-specific MFM is to attempt to capture additional noise and errors outside of the measurement error. In Sections 3–5, we saw that using the circuit-specific MFM to post-process a measured distribution often led to lower values of the Kullback-Leibler metric compared to the more general, hardware MFM and lower MMD values during training.

In the main text, we discussed circuits with sparse entangling layers, trained on hardware with connectivity that supports the gate layout. The d_C = 3 circuit was trained on 3-qubit devices (ibmq_boeblingen, ibmq_valencia, ibmq_20_tokyo) but the d_C = 2 circuit was only trained on ibmq_20_tokyo. The sparsity of the entangler layers and the optimization with hardware layout is designed to reduce the overall amount of 2-qubit gate noise by reducing the number of noisy gates in a compiled circuit. The compiled circuits executed on hardware in the main text had the minimum number of CNOTs (4 and 6, for d_C = 2 and d_C = 3 respectively).

If a circuit is executed on hardware that does not exactly match the hardware coupler layout, then additional CNOTs will be added to the compiled circuit. We tested the efficacy of EM for noise associated with added CNOTs by running the same experiments from Section 4 using a d_C = 2 circuit on ibmq_boeblingen. On ibmq_20_tokyo, the compiled circuit contained only 4 CNOT gates, but on ibmq_boeblingen. the compiled circuit could have up to 16 CNOT gates. The d_C = 2 circuit was trained for 20 steps of Adam with BAS(2,2) as the target.

Now, circuits trained with $\mathbb {K}_{hw}$ or $\mathbb {K}_{circ}$ in training shows no substantial improvement in the MMD loss minimization, and in Fig. 13, we see that the inclusion of EM in training can significantly impair the minimization of ${{\mathcal{L}}_{MMD}}$. The $\min \limits {({\mathcal{L}}_{MMD})}$ measured was 0.0332, 0.033, and 0.0281 for , and $\mathbb {K}_{circ}$, respectively, which is far higher than the minimum value found during noiseless simulation $\min \limits {[{\mathcal{L}}_{MMD}])} = 0.0045$. Yet post-processing did show improvement in the Kullback-Leibler metric (reported in Table 9). In the main text, we observed a lowest value of $\min \limits {\langle D(P|Q) \rangle }= 0.0763$ for a d_C = 2 circuit trained on ibmq_20_tokyo. Now on ibmq_boeblingen with additional CNOTs the lowest value of $\min \limits {\langle D(P|Q) \rangle }= 0.1490$. Figure 13 shows interesting behavior for the value of $\min \limits {\langle D(P|Q) \rangle }$ post-processed with EM. At the end of the training, the metric is significantly higher than the noiseless simulation, but there are points where the KL value post-processed with $\mathbb {K}_{circ}$ closely matches the noiseless simulation. A simple explanation for this discrepancy is that the additional CNOT noise impairs the gradient-based training causing it to leave a local minima at step 16. Our EM method cannot compensate for this significantly higher level of gate noise associated with the additional CNOT. This was also observed in the main text (see Fig. 6)—if the training gets trapped in a poor local minima this cannot be corrected simply through post-processing (Fig. 14).

Fig. 13

${{\mathcal{L}}_{MMD}}$ for a d_C = 2 circuit during 20 steps of training with Adam on ibmq_boeblingen, with BAS(2,2) as target and 3 MFM in training. (left) Measured on T₀ for (black, circles, dotted), $\mathbb {K}_{hw}$ (red, squares, solid), and $\mathbb {K}_{circ}$ (blue, diamonds, dashed). (right) Measured on T₁ for (black, circles, dotted), $\mathbb {K}_{hw}$ (red, squares, solid), and $\mathbb {K}_{circ}$ (blue, diamonds, dashed)

Table 9 $\min \limits {\langle D(P|Q)\rangle }$ values observed in post-processed data for d_C = 2 circuits trained on ibmq_boeblingen

Fig. 14

〈D(P|Q)〉 for d_C = 2 circuit trained on ibmq_boeblingen with BAS(2,2) as target and 20 steps of Adam with identity-MFM in training. Averaged over 10 sub-samples of (left) 4096 shots, (center) 2048 shots, (right) 512 shots. The scores are post-processed with: no error mitigation (black, circles), hardware-MFM mitigation (red, square) and circuit-MFM mitigation (blue, diamonds)

Appendix B: Measurement fidelity matrix construction

Our approach to error mitigation is a generalization of the measurement fidelity matrices used in previous studies (Bialczak et al. 2010; Kandala et al. 2017), but instead of constructing the matrix from 1-qubit errors, we extract the error correlations from the output of multi-qubit circuits executed on localized qubit subsets. For a system of N qubits, a set of 2^N quantum circuits is used to prepare each individual state of the computational basis.

$\mathbb {K}_{hw}$ only mitigate readout errors and is generated row-wise using shallow circuits of N qubits and up to N individual X gates. We extended this MFM construction method to a more general structure so we can estimate the general error associated with a particular parameterized circuit. We constructed the matrices $\mathbb {K}_{circ}$ used in the main text using the general circuit ansatz of Fig. 1. Example of the circuits used to construct $\mathbb {K}_{hw}$ and $\mathbb {K}_{circ}$ are shown in Fig. 15.

Fig. 15

(Left) Example of a circuit used to construct a row of $\mathbb {K}_{hw}$. (Right) Example of a circuit used to construct a row of $\mathbb {K}_{circ}$

Since we project the final state prepared by the circuit onto a set of real-valued amplitudes, there are multiple sets of rotational parameters that will result in the same final distribution. The rotational parameters in Rotation Layer 1 and Rotation Layer 3 are all set to 𝜃 = 0 such that both layers are effectively implementing identity gates. Each qubit in Rotation Layer 2 has a gate sequence R_Z(𝜃_i)R_X(𝜃_j)R_Z(𝜃_k) applied to it. By replacing this gate sequence with either R_Z(0)R_X(0)R_Z(0) or R_Z(−π/2)R_X(−π)R_Z(−π/2), we can use the parameterized ansatz to prepare any basis state. For example, applying R_Z(0)R_X(0)R_Z(0) to all 4 qubits will return only the state |0000〉 in the absence of noise.

When deployed on noisy hardware, we can again construct an MFM that quantifies the probability of the target state being prepared, and the probabilities of other states being prepared. In Section 5, we calculated the Frobenius norm of the difference matrix and showed that $\mathbb {K}_{circ}$ matrices are much farther from the identity matrix than $\mathbb {K}_{hw}$ matrices. The final MFM is assumed to be invertible and non-negative, but we note that the matrix is not symmetric. Additionally, the individual state fidelities are not uniform. For 3 MFM generated on the same day on the same set of qubits, we report the individual basis state fidelities in Table 10.

Table 10 Individual state fidelities for d_C = 2, d_C = 3 $\mathbb {K}_{circ}$ and $\mathbb {K}_{hw}$ all evaluated at P_B on ibmq_20_tokyo