SchWARMA: A model-based approach for time-correlated noise in quantum circuits

Temporal noise correlations are ubiquitous in quantum systems, yet often neglected in the analysis of quantum circuits due to the complexity required to accurately characterize and model them. Autoregressive moving average (ARMA) models are a well-known technique from time series analysis that model time correlations in data. By identifying the space of completely positive trace preserving (CPTP) quantum operations with a particular matrix manifold, we generalize ARMA models to the space of CPTP maps to parametrize and simulate temporally correlated noise in quantum circuits. This approach, denoted Schrödinger wave ARMA (SchWARMA), provides a natural path for generalization of classic techniques from signal processing, control theory, and system identification for which ARMA models and linear systems are essential. This enables the broad theory of classical signal processing to be applied to quantum system simulation, characterization, and noise mitigation.

Figure 1

Circuit schematic demonstrating how one might implement a SchWARMA model within a circuit of arbitrary one- and two-qubit gates labeled with $G_i$. It allows one to consider perfect gates followed by noise that retains the time correlation. The $S_k^{\left(q\right)}$ terms are the error channel defined in Eq. (5b) with additional upper index denoting the qubit.

Figure 2

Simulated QNS experiments. (a) Spectrum reconstructions (markers) of SchWARMA simulated low-pass and band-pass noise (lines). (b) Spectrum reconstruction of SchWARMA simulated “peaky” multipole AR spectrum. (c) Spectrum reconstructions of SchWARMA simulated $f^{-\alpha }$ noise; dashed lines indicate ideal $\alpha$.

Figure 3

SchWARMA control simulations. (a) Process fidelity decay due to multiaxis coherent noise. Correlated noise $({\tau }_c=3)$ produces more rapid decay than uncorrelated noise. Decoupling protocols reduce the decay for correlated case, with slower decay for $XY4$ than $XX$. (b) Decay in unitality from amplitude damping. Uncontrolled decay is essentially identical, but decoupling protocols better preserve unitality for correlated damping (${\tau }_c=3$). In this case, $XX$ and $XY4$ (not shown) are identical.

Figure 4

The $X$ stabilizer is given on the left and the $Z$ stabilizer is given on the right. Measurements of the ancilla are omitted from these circuits.

Figure 5

The CNOT decomposed into elementary gates. The $Z$ rotations are taken to be virtual within all of our simulations, and take no time and occur without error.

Figure 6

These two plots provide an illustrative example of how the full simulation of the surface code circuits are done. In (a), we show the single-qubit control amplitudes, $\mathrm{\Omega }\left(t\right)$, for the Hamiltonian in Eq. (23) and the two-qubit control amplitudes, $A\left(t\right)$, for the Hamiltonian in Eq. (24) for the qubits labeled in the legend. We see single-qubit $X$ and $Y$ controls being applied at the beginning and end of the circuit. These correspond to the application of the Hadamard gate for the $X$ stabilizer. In the middle, the entangling operations are performed. In (b), we plot a single realization of a noise trajectory, ${\eta }_i\left(t\right)$, for the multiaxis noise term in Eq. (23) for the same qubit used in (a). We choose noise values ${\gamma }_x={\gamma }_y={\gamma }_z={10}^{-4}$ and the correlation time ${\tau }_{c,x}={\tau }_{c,y}={\tau }_{c,z}=32$. Those are in units of inverse gate length and gate length, respectively. The noise is always on across the entire duration of the circuit. These values yield a highly time-correlated trajectory.

Figure 7

Process infidelities for surface code $Z$ check circuits (a) and the absolute error between the SchWARMA and Trotter simulations (b) as a function of noise parameters ${\tau }_{c,i}$ [(a) $x$ axis], ${\gamma }_i$ (color) defined below Eq. (21), and process infidelity [(b) $x$ axis]. The noise strengths are ${\gamma }_{x,y,z}\in \{{10}^{-12},{10}^{-10},...,{10}^{-2}\}$ [bottom to top on (a) plot] in units of inverse gate length. In (a), the Trotter simulations are the line plot while the SchWARMA simulations are the marks with Monte Carlo (1000 samples) error bars. In (b), the absolute difference between the SchWARMA and Trotter simulations is plotted with a fit line that is consistent with Monte Carlo sampling error.

Figure 8

SchWARMA (symbols) vs analytical results and full dynamics simulation of continuously driven systems (both shown as solid lines). The comparison indicates good agreement between SchWARMA and alternative approaches. LZ results are shown for an (a) 2-level and (b) 3-level system with noise power $f\left(0\right)=0.003\alpha$ for different driving rates $\alpha$. Grover's search algorithm comparisons [panel (c)] focus on varying correlation time ${\tau }_c$ using a fixed noise power of $f\left(0\right)=0.001{\mathrm{\Delta }}_{min}^2$.

Figure 9

Monte Carlo sampling error analysis. (a) Plots of the absolute error between two SchWARMA Monte Carlo samples with identical theoretical infidelities for a range of infidelities and sample sizes. Note the considerable variation and overlap between the data corresponding to different sample sizes, and the similarity to Fig. 3, right panel. The dashed lines indicate the expected error based on the sample size. (b) Plot showing the required sample size needed for a relative error to achieve a statistically significant result using a single-sided $F$ test at the $p=0.05$ level. Note the slope of this line is $-2$, in line with the intuition that one needs samples proportional to the square of the sensitivity of the test.