Theory of quantum error mitigation for non-Clifford gates

TL;DR

This work tackles error mitigation for non-Clifford gates, addressing a key barrier to using error-mitigation techniques in semi-analog quantum simulations that rely on weakly-entangling, non-Clifford gates. It introduces Pauli shaping, a general framework that maps a noisy gate to a desired channel via a linear combination of Pauli flips, enabling noise cancellation or amplification with a sampling overhead; this generalizes probabilistic error cancellation and zero-noise extrapolation beyond Clifford gates. A second pillar develops SPAM-robust noise-learning methods for common non-Clifford two-qubit gates, focusing on $R_{ZZ}( heta)$, including three learning schemes that recover relevant PTM elements with controlled overhead and concentration properties. The paper also identifies a fundamental trade-off: non-Clifford noise can be more complex to mitigate than Clifford noise, with potential discontinuities in overhead and the appearance of hard-to-learn Type 4 PTM elements, highlighting directions for future work in scalable learning and gate-engineering strategies. Overall, the results illuminate when and how non-Clifford error mitigation might be advantageous and establish practical pathways for extending PEC/ZNE-style mitigation to a broader class of quantum gates.

Abstract

Quantum error mitigation techniques mimic noiseless quantum circuits by running several related noisy circuits and combining their outputs in particular ways. How well such techniques work is thought to depend strongly on how noisy the underlying gates are. Weakly-entangling gates, like $R_{ZZ}(θ)$ for small angles $θ$, can be much less noisy than entangling Clifford gates, like CNOT and CZ, and they arise naturally in circuits used to simulate quantum dynamics. However, such weakly-entangling gates are non-Clifford, and are therefore incompatible with two of the most prominent error mitigation techniques to date: probabilistic error cancellation (PEC) and the related form of zero-noise extrapolation (ZNE). This paper generalizes these techniques to non-Clifford gates, and comprises two complementary parts. The first part shows how to effectively transform any given quantum channel into (almost) any desired channel, at the cost of a sampling overhead, by adding random Pauli gates and processing the measurement outcomes. This enables us to cancel or properly amplify noise in non-Clifford gates, provided we can first characterize such gates in detail. The second part therefore introduces techniques to do so for noisy $R_{ZZ}(θ)$ gates. These techniques are robust to state preparation and measurement (SPAM) errors, and exhibit concentration and sensitivity--crucial statistical properties for many experiments. They are related to randomized benchmarking, and may also be of interest beyond the context of error mitigation. We find that while non-Clifford gates can be less noisy than related Cliffords, their noise is fundamentally more complex, which can lead to surprising and sometimes unwanted effects in error mitigation. Whether this trade-off can be broadly advantageous remains to be seen.

Figures (13)

• Figure 1: The two conceptual steps of PEC, and the related form of ZNE, for a noisy Clifford gate $\mathcal{G}$. Any $\mathcal{G}$ can described as a noise channel $\mathcal{N}$ followed by an ideal gate $U$. (We write the unitary $U$ in place of the channel $\mathcal{U}$ in the circuit above, and likewise for other noiseless gates, for simplicity.) In Step 1, one adds random Pauli gates on both sides of $\mathcal{G}$, chosen so as to twirl $\mathcal{N}$ into a Pauli channel $\bar{\mathcal{N}}$. We denote the resulting average channel as $\bar{\mathcal{G}}$. In Step 2, one then adds a random Pauli gate $P_k$ before $\bar{\mathcal{G}}$, sampled from a probability (for ZNE) or quasi-probability (for PEC) distribution $\vec{q}$, which is chosen to correctly amplify ($\alpha>0$) or invert ($\alpha=-1$) $\bar{\mathcal{N}}$, respectively. We denote the resulting aggregate channel as $\mathcal{A}$, and use the notation $\stackrel{\text{avg}}{=}$ to indicated that, for PEC, the relation only holds for expectation values.
• Figure 2: A thought-experiment where a noisy non-Clifford gate $\mathcal{G}$ happens to factorize into an ideal gate $U$ followed by a Pauli noise channel $\mathcal{N}'$, as in the middle circuit. $\mathcal{N}'$ can therefore be amplified or inverted by inserting random Paulis after the noisy gate. However, if we factorized $\mathcal{G}$ as shown on the right, the resulting noise channel $\mathcal{N}$ would generally be non-Pauli, and could not be amplified or inverted by inserting random Paulis before $\mathcal{G}$. This is a fundamental difference between Clifford and non-Clifford gates.
• Figure 3: A circuit description of Pauli shaping. To shape a given channel $\mathcal{G}$ into a desired channel $\mathcal{A}$, one can insert random $n$-qubit Paulis $P_j$ and $P_i$ before and after $\mathcal{G}$, respectively, to form a channel $\mathcal{A}^{(ij)}$. One picks $P_i$ and $P_j$ randomly with probability $|\boldsymbol{Q}_{ij}|/\gamma$ in each shot, then multiplies the measurement outcomes by $\gamma \mathop{\mathrm{sgn}}\nolimits(\boldsymbol{Q}_{ij})$ to realize $\mathcal{A}$, for $\boldsymbol{Q}$ and $\gamma$ from Eqs. \ref{['eq:pauli_shaping']} and \ref{['eq:pauli_shaping_gamma']} respectively. When all $\boldsymbol{Q}_{ij} \ge 0$, i.e., when $\boldsymbol{Q}$ is a valid probability distribution of pairs of Paulis, the last step is trivial and the channel $\mathcal{A}$ is actually realized. When some $\boldsymbol{Q}_{ij} < 0$, $\boldsymbol{Q}$ is instead a quasi-probability distribution, and $\mathcal{A}$ is only realized in terms of expectation values, as indicated by the notation $\stackrel{\text{avg}}{=}$.
• Figure 4: Two equivalent ways to view Clifford PEC/ZNE. Left: the conceptual steps are shown separately, where the random Paulis $P_i\sim \text{unif}(\mathbb{P})$ and $P_j = P_{\sigma(i)} \propto U^\dag P_i U$ serve to twirl the factored noise channel $\mathcal{N}$, then a random Pauli $P_k$ is added with quasi-probability $k \sim \vec{q}$ to amplify or invert the twirled noise. Right: $P_k$ and $P_j$ are combined into a single Pauli $P_\ell = P_{k \oplus \sigma(i)}$, so $P_\ell$ and $P_i$ are added before and after $\mathcal{G}$, respectively, with quasi-probability $q_k / |\mathbb{P}| = 4^{-n} \, q_{\sigma(i) \oplus \ell}$.
• Figure 5: A noisy $R_{ZZ}(\theta)$ gate used as a numerical example. Left: The PTM of the channel $\mathcal{G}$ constructed in Example 3 of Sec. \ref{['sec:examples']}, which describes a noisy $U= R_{ZZ}(47^\circ)$ gate. Center: The PTM of the associated noise channel $\mathcal{N} = \mathcal{U}^{-1} \mathcal{G}$, with the identity matrix subtracted to reduce the range of values and thereby improve visibility. Right: The relative error in learning the PTM elements of this $\mathcal{G}$ that are essential for Pauli shaping, using the schemes developed in Sec. \ref{['sec:learning']}. The learned elements are all accurate to within $\lesssim 1\%$ (with room for further optimization), even though the underlying simulation includes state preparation and measurement (SPAM) errors, each with only 90% average fidelity nielsen:2002. The elements in gray were not learned, as their values are not needed to cancel the noise through Pauli shaping, which requires one to find numbers $\{ \boldsymbol{C}_{ij} \}_{i,j=0}^{15}$ such that $\boldsymbol{U}_{ij} = \boldsymbol{C}_{ij} \boldsymbol{G}_{ij}$ (as in Eq. \ref{['eq:pauli_shaping']}, with $\boldsymbol{A}=\boldsymbol{U}$). There is no need to learn $\boldsymbol{G}_{0,0}$, as it equals 1 for any CPTP map, which fixes $\boldsymbol{C}_{0,0}=1$. For all other grayed out indices, $\boldsymbol{U}_{ij}=0$ (see Eqs. \ref{['eq:U_PTM']} and \ref{['eq:R2']}), so can we can pick $\boldsymbol{C}_{ij}=0$ regardless of $\boldsymbol{G}_{ij}$.