Noise Correlations as a Resource in Pauli-Twirled Circuits

Abstract

Randomized compiling (RC) is an established tool to tailor arbitrary quantum noise channels into Pauli errors. The effect of both spatial and temporal noise correlations in randomly compiled circuits, however, is not fully understood. Here, we show that for a broad class of correlated Gaussian noise, RC reduces both the strength and temporal range of correlations. For Clifford circuits, we derive a simple analytical expression for the circuit fidelity of randomly compiled circuits. Surprisingly, we show that this fidelity is always increased by the presence of correlations, suggesting that correlations are a resource in randomly compiled circuits. To leading order in system-bath coupling, we also show that RC suppresses the quantum component of bath correlations, implying that one can safely treat weak noise as being classical. Finally, through extensive numerical simulations, we show that our results remain valid for many relevant non-Clifford circuits. These results clarify how RC mitigates memory effects and enhances circuit robustness.

Figures (5)

• Figure 1: (a) A quantum circuit composed of layers of Clifford gates (yellow rectangle) subject to spatio-temporally correlated noise (green boxes with dotted lines). We consider two noise models, correlated dephasing along the $Z$ axis and correlated Pauli channels [c.f. Eq. \ref{['overrotations']} and Eq. \ref{['pauli_layer']}]. (b) Circuit fidelity of bare and twirled Clifford circuits as a function the correlation time of the noise. Noise temporal correlations are decaying exponentially with correlation time $\tau_c$ and is spatially uncorrelated [c.f. Eq. \ref{['corr_func']}]. Each blue curve corresponds to a different bare Clifford circuit fidelity. The fidelity of each of these circuits is equivalent after twirling the noise (red curve). The twirled circuit fidelity increases with the correlation time of the noise.
• Figure 2: Noise-averaged circuit fidelity of random Clifford and non-Clifford brickwork circuits under non-Markovian $Z$-dephasing, comparing the bare noise model to its Pauli-twirled effective description. (a) Circuit construction. We generate $20$ random depth-$16$ circuits on $16$ qubits composed of alternating layers of nearest-neighbor CNOT gates and single-qubit layers in which a gate is applied independently to each qubit with probability $0.3$, chosen to be $S$ or $\sqrt{X}$ with equal probability. Non-Clifford circuits are obtained by additionally inserting a $T$ gate independently for each qubit with probability $0.5$. (b) Circuit fidelity as a function of noise correlation time. The dephasing noise has covariance given by Eq. \ref{['corr_func']} with $\sigma=0.15$; data are shown for correlation times $\tau/t_g = 0.1, 1, 10,$ and $100$. Error bars from noise averaging are smaller than the marker size, so the visible spread reflects circuit-to-circuit variation. Under Pauli-twirling, fidelities are strongly concentrated across circuit instances, whereas in the bare model substantial circuit-to-circuit variation is observed.
• Figure 3: Circuit fidelity as a function of the noise correlation time. We simulate the circuit fidelity of the qft_n_18.qasm (red) and sqrt_n_18.qasm (blue) circuits subject to non-Markovian dephasing noise with covariance given by Eq. \ref{['corr_func']} and noise strength $\sigma = 0.035$. For each circuit, we compare the bare implementation (circles) to its Pauli-twirled counterpart (triangles). Error bars corresponding to the standard deviation with respect to the noise averaging were calculated but are too small to be noticeable. In the Pauli-twirled case, the state fidelity exhibits only a weak dependence on the correlation time, whereas in the bare case it varies strongly as temporal correlations increase.
• Figure 4: Logical survival probability of the $[[3,1,1]]$ repetition code initialized in the logical $\ket 0$ state after $250$ error-correction cycles as a function of the noise correlation time. Each data and ancilla qubit is subject to non-Markovian dephasing noise after each two-qubit gate, with correlation function Eq.\ref{['corr_func']} and noise strength $\sigma = 0.05$. We compare the Pauli-twirled (green) and bare (orange) circuit implementations. Error bars corresponding to the standard deviation with respect to the noise averaging were calculated but are too small to be noticeable. In the Pauli-twirled case, the logical survival probability increases slightly with correlation time, whereas in the bare case it exhibits a pronounced non-monotonic dependence.
• Figure 5: Circuit fidelity for $100$ random Clifford and non-Clifford circuit. We consider the Pauli-twirled and bare circuit fidelities. The number of qubits is $16$ and the depth is $16$ two-qubit gate layers. The circuits are generated using the same procedure as in Fig. \ref{['fig:variance_fig']} with the addition that uniformly random single qubit Clifford gates are applied between each two-qubit gate layers. Each circuit is subject to a noise with a covariance matrix Eq. \ref{['corr_func']} with correlation time is $\tau = 100t_G$ and $\sigma = 0.035$. We see that twirling reduces the variance, but does not change the average in this case.