Disambiguating Pauli noise in quantum computers

Abstract

To successfully perform quantum computations, it is often necessary to first accurately characterize the noise in the underlying hardware. However, it is well known that fundamental limitations prevent the unique identification of the noise. This raises the question of whether these limitations impact the ability to predict noisy dynamics and mitigate errors. Here, we show, both theoretically and experimentally, that when learnable parameters are self-consistently characterized, the unlearnable (gauge) degrees of freedom do not impact predictions of noisy dynamics or error mitigation. We use the recently introduced framework of gate set Pauli noise learning to efficiently and self-consistently characterize and mitigate noise of a complete gate set, including state preparation, measurements, single-qubit gates and multi-qubit entangling Clifford gates. We validate our approach through experiments with up to 92 qubits and show that while the gauge choice does not affect error-mitigated observable values, optimizing it reduces sampling overhead. Our findings address an outstanding issue involving the ambiguities in characterizing and mitigating quantum noise.

Figures (15)

• Figure 1: Overview of results. Leading error mitigation methods based on Pauli noise models presuppose accurate knowledge of all the hardware error rates. This foundational assumption, however, is false, as it has been proven that such a noise model cannot be uniquely determined by experiments, even in principle. Without accounting for this indeterminacy, previous experiments implicitly used an inconsistent set of the gauge parameters $\{\mathcal{D}_{\boldsymbol{\eta}_1}, \mathcal{D}_{\boldsymbol{\eta}_2} \}$ across the quantum gate set, e.g., different gauge choices for state preparation and measurement and the two-qubit gate (top, blue). We show that a self-consistent set of gauge parameters (middle, pink) is necessary for unbiased quantum error mitigation, as exemplified here in the mitigation bias of state preparation experiment of an $n=21$ entangled state known as the Greenberger–Horne–Zeilinger state (GHZ) state (upper right; details in Fig. \ref{['fig:6']}). Furthermore, the choice of a consistent gauge can be optimized (bottom, green) to reduce the sampling overhead of error mitigation (bottom, right).
• Figure 2: Pattern transfer graph. Example of a $2$-qubit gate set that contains CNOT as the only entangling gate, where any observable can be mapped to a cycle starting from the State-preparation and Measurement (SM) node and back. The bi-directional gray dashed line has 18 SPAM eigenvalues associated with it that have been omitted for clarity. The numbers in the nodes denote patterns of Pauli operators, i.e., a bit string indicating which qubits have a nontrivial single-qubit Pauli operator supported on them.
• Figure 3: Quasi-local noise model. An example of a $3$-qubit system with $2$-local noise generators.
• Figure 4: Graphical proof that a complete gate set learned in a self-consistent manner can be validated using an error mitigation formalism. (a) Using the probabilistic error cancellation (PEC) framework, the quasi-inverse channels $\Lambda_{S/\mathcal{G}/M,\bm \eta}^{-1}$ can be applied at the expense of a sampling overhead which increases exponentially with the amount of noise in the constituent noise channels. Unlike previous formulations, we apply this inverse channel in a self-consistent manner where the entire gate set shares the same set of gauge parameters $\bm \eta$. For the sake of clarity, we chose a controlled-NOT gate as the two-qubit gate. This same proof applies to any other type of two-qubit gate, and to arbitrary numbers of qubits. (b) Substitution of the gauge $\mathcal{D}_{\bm\eta}$ and Pauli noise channels $\Lambda_{S/\mathcal{G}/M}^{-1}$ from Fig. \ref{['fig:1']}b yields the operations in the red boxes. (c) Reordering the gauge channels (pink arrows) leads to cancellations, or compositions of identity channels (pink brackets). Note that Pauli channels commute with each other. (d) The gauge channels and their inverses also compose to identity channels (pink brackets). Note that the generalized depolarizing channels commute with any single-qubit gates (gray squares). Also recall $\mathcal{D}_{\bm\eta}^{'}$ is defined to be the gauge channel conjugated by the entangling Clifford gate ($\mathcal{G} \circ \mathcal{D}_{\bm\eta}^{'} = \mathcal{D}_{\bm\eta} \circ \mathcal{G}$). (e) Finally, the resulting, mitigated circuit shows a noise-free operation of a controlled-NOT gate on two qubits.
• Figure 5: Experimental learning and mitigation of a restricted set of errors on two qubits. (a) Circuit used for both learning and targeted mitigation, except an initial state with $m_1=m_2=0$ is used for the learning while $m_1=m_2=1$ is used for the target circuit. This restricts learning to three Pauli terms: $IZ$, $ZZ$ and $ZI$. Note that the target circuit is prepared in $|11\rangle$, the $-IZ$ and $+ZZ$ eigenstates, which differs from the learning circuit which is prepared in $|00\rangle$, the $+IZ$ and $+ZZ$ eigenstates. The "inconsistent" noise model only requires learning with circuits using even depths (0, 2, 4,.. 32), while the "consistent" learning model requires one additional depth-1 experiment. (b, c) For the non-degenerate (b) and degenerate (c) cycles, the experimental data (gray) from the target circuit is shown alongside the predicted outcomes using the inconsistent (blue) and consistent (pink) noise models. To the right of (b), both the even and odd depths show no difference in predicted outcomes. However, to the right of (c), the even depth shows no difference while the odd depths show a difference of 3.2$\pm$0.4% (blue, inconsistent) compared to a 0.5$\pm$0.3% (pink, consistent) bias in the outcomes.

Theorems & Definitions (3)

• Theorem 1: Self-consistent PEC
• Proposition 2
• Proposition 3: Theorem 1 in main text