Towards Predictive Quantum Algorithmic Performance: Modeling Time-Correlated Noise at Scale

TL;DR

Combining tensor network techniques with quantum autoregressive moving average models, this work quantify the effects of time-correlated noise on quantum algorithms and predict their performance at scale, paving the way for large-scale algorithmic simulations and performance prediction under hardware-relevant noise conditions informed by realistic device characteristics.

Abstract

Combining tensor network techniques with quantum autoregressive moving average models, we quantify the effects of time-correlated noise on quantum algorithms and predict their performance at scale. As a paradigmatic test case, we examine the quantum Fourier transformation. Building on our first technical result, which shows how stochastic tensor network calculations capture frequency correlations, our second result is the revelation that infidelity exponents (scaling from diffuse, to superdiffuse) are determined by the spectral features of the noise. This numerical result rigorously quantifies the common belief that the temporal correlation scale is a key predictive feature of noise's deleterious impact on multi-qubit circuits. To highlight prospects for predicting algorithmic performance, our third result quantifies how infidelity scaling exponents -- which are fits determined by training data at moderate scales (40-80 qubits) -- can be used to predict more computationally expensive simulation at larger scales (100-128 qubits). Aside from highlighting the scalability of our methods, this workflow feeds into our last result, which is the proposal of predictive benchmarking protocols connecting simulations to experiments. Our work paves the way for large-scale algorithmic simulations and performance prediction under hardware-relevant noise conditions informed by realistic device characteristics.

Figures (4)

• Figure 1: Graphical circuit representation of an $N=3$ qubit noisy QFT. The QFT circuit (black) is deformed with time-correlated dephasing noise (red gates). The $y^{(l)}_{(q,i)}$s ($q^\text{th}$ qubit, $i^{\text{th}}$ time step and $l^{\text{th}}$ instance of the ARMA model) are sampled from a physically relevant spectrum via the SchWARMA formalism. The $y^{(l)}_{i}$s on the same qubit at different time steps are correlated (time correlated) while $y^{(l)}_{i}$s on different qubits at the same time step remain uncorrelated (spatially uncorrelated) and are sampled from independent SchWARMA processes.
• Figure 2: Infidelity, $\mathcal{I}$ as a function of total noise power, $P_{tot}$ for the different regimes of (top) $\sigma \gg \alpha$, (bottom) $\alpha \gg \sigma$, for a system size of $N=40$ qubits. For the input state, $\ket{\eta}$, we consider a random MPS characterized by maximum bond dimension, $\chi=4$. In both regimes, we fit the $\mathcal{I}$, to a power law scaling i.e., $\mathcal{I} = \lambda P_{tot}^{\xi}$ where the exponents are given by (top) $\xi = 0.98 \pm 0.06$ (bottom), $\xi = 1.38 \pm 0.08$.
• Figure 3: (a) $\mathcal{I}$ as a function of total noise power, $P_{tot}$ and depth, $D$ in the regime of $\sigma \gg \alpha$. For system sizes ($40 \le N \le 80$), we set the input state, $\ket{\eta}$, to be a random MPS characterized by maximum bond dimension, $\chi=4$. Following the earlier results, to extract the scaling behavior, we perform a power-law fit of infidelity $\mathcal{I}$ as a function of the $P_{tot}$ and $D$, with the fit function given by $\mathcal{I} = \Lambda P_{tot}^{\Xi}D^{\Upsilon}$, where $\Xi = 0.965 \pm 0.013, \Upsilon = 0.958 \pm 0.035$ are the fit parameters for the exponents. (b) The power law fit is used to predict the infidelity, $\mathcal{I}$ (denoted by the orange line) for a larger system size of $N=100$ qubits. The fit captures the trend in the simulation data (denoted by the blue dots) particularly well for low total noise power.
• Figure 4: Probability of bistrings sampled from a ensemble of noisy states obtained by the application of $\widetilde{\text{QFT}}(N) \text{QFT}(N)$ on a $N=128$-qubit product state for different total noise power, $P_{tot}$ injected into the circuit. We estimate the total noise power $P_{tot} = P_{0}$, using the power law scaling, $\mathcal{I} = \tilde{\Lambda} P_{0}^{\tilde{\Xi}}D^{\tilde{\Upsilon}}$ by fixing the infidelity, $\mathcal{I}$ to be 15%. The number of bitstring samples is set to $10^4$ for each independent noisy trajectory, with the total number of noisy trajectories set to $10^3$ resulting in a total number of $10^7$ bitstring samples. (Inset) A magnified view of the first 30 high probability bit strings excluding the probability of the initial state. With the increase in power, the return probability to the initial state ($1 - \mathcal{F}$) reduces, with the leakage in probability extending to other bitstrings. At higher powers, the leakage in return probability remains higher in comparison to lower powers thereby reflecting the behavior of the corresponding infidelity.