Modeling noisy quantum circuits using experimental characterization

Noisy intermediate-scale quantum (NISQ) devices offer unique platforms to test and evaluate the behavior of quantum computing. However, validating circuits on NISQ devices is difficult due to fluctuations in the underlying noise sources and other nonreproducible behaviors that generate computational errors. Here we present a test-driven approach that decomposes a noisy, application-specific circuit into a series of bootstrapped experiments on a NISQ device. By characterizing individual subcircuits, we generate a composite noise model for the original quantum circuit. We demonstrate this approach to model applications of Greenberger-Horne-Zeilinger(GHZ)-state preparation and the Bernstein-Vazirani algorithm on a family of superconducting transmon devices. We measure the model accuracy using the total variation distance between predicted and experimental results, and we demonstrate that the composite model works well across multiple circuit instances. Our approach is shown to be computationally efficient and offers a trade-off in model complexity that can be tailored to the desired predictive accuracy.

Figure 1

An example of a subcircuit decomposition where subcircuit set $S=\{S_{\text{blue}},S_{\text{green}}\}$.

Figure 2

A graphical representation of the register connectivity in the Poughkeepsie QPU at the time of data collection, in which each node corresponds to a register element and directional edges indicate the availability of a programmable two-qubit cross-resonance gate.

Figure 3

Schematic representation of the quantum circuit used for preparation of the $n$-qubit GHZ state defined by Eq. (2). The circuit layout satisfies the connectivity constraints of the IBM Poughkeepsie QPU shown in Fig. 2. The circuit uses a total of $n-1$ cnot gates and $n$ measurement gates. Colored boxes denote subcircuit selections.

Figure 4

The test circuit for characterizing the cnot operation acting on register elements $q_j$ and $q_k$. This test prepares the two-qubit Bell state as an instance of $n=2$ in Fig. 3.

Figure 5

Error rates under the ARO channel for each qubit of Poughkeepsie. The SRO channel is given by the error rates for state 0 shown here. Average $p_0$ value is 0.0212 (standard deviation of 0.0101 across all qubits) and average $p_1$ value is 0.0681 (standard deviation of 0.0233). Each qubit is evaluated in a separate circuit, e.g., $X_0\left|0_0,0_1,...,0_{19}\right.〉$ to generate Eq. (6) for qubit 0.

Figure 6

Depolarizing error rates associated with $X$ gate application for each qubit of Poughkeepsie. Average $p_x$ value is 0.0033 with standard deviation 0.00303.

Figure 7

Error rates for cnot gates under the depolarizing channel for each coupled qubit pair of Poughkeepsie. These values are fitted to include the ARO channel noise with rates shown in Fig. 5. Reported error bars represent the upper limit of the error from the least-squares calculation.

Figure 8

Comparison of possible choices for the composite model. The best performance is achieved in the $\mathrm{ARO}+\mathrm{DP}$ case. Error bars represent the distribution of TVD values across 100 sets of 8192 samples per simulation case.

Figure 9

Performance of the selected noise model on $n$-qubit GHZ states. The best performance is achieved with the fully spatial noise model. Error bars represent the distribution of TVD values across six sets of 8192 samples per simulation case.

Figure 10

Scaled performance of selected noise model on $n$-qubit GHZ states, where TVD is divided by the number of cnot gates in each circuit. Error bars represent the distribution of TVD values across six sets of 8192 samples per simulation case.

Figure 11

Circuit implementation of the Bernstein-Vazirani algorithm. The bottom qubit of the register is the oracle; the top three yield the secret string, here given as 101 as an example. Other secret strings are produced by changing the cnot gate sequence such that control qubits correspond to output bits of 1.

Figure 12

Performance of Bernstein-Vazirani algorithm evaluated as the measured probability of the prepared secret string. Simulation is subject to noise defined by the fully spatial model.