Mathematical formalism

Given the task of learning Pauli noise for a set of Clifford gates, a common analysis technique is to track how individual Pauli operators are transformed through a sequence of gate applications (under the assumption that the gates are Clifford and the noise is a Pauli channel, a single Pauli will always be mapped to another Pauli up to a scalar that is a function of the noise). Obviously, the evolution of any state under such a sequence can be captured by a linear combination of such trajectories, but under the assumption that we always prepare the state in a Pauli eigenstate, and always measure and post-process the results to compute the expectation value of a Pauli operator, the relationship between the expectation value and the noise model parameters will always depend on only a single such trajectory. This follows from the simple fact that the initial state is a linear combination of Pauli operators, each gate and noise model maps Paulis to Paulis and preserves their orthogonality, and the final expectation value “selects” only one of the Paulis in the final decomposition before measurement.

This type of reasoning appears in many parallel research tracks in noise learning, including in the Pauli gate-set learning literature [3], [4], the ACES literature [6], and the cycle benchmarking literature [5]. This reasoning was further generalized in [7] to include Clifford-MCM gates (a Clifford gate followed by a projective mid-circuit measurement). This package most closely follows [4], [7]. While not explicitly named, we adopt the Pattern Transfer Graph (PTG) formalism for describing how Pauli operators evolve through learning circuits, providing a direct data representation of paths through the graph.

The following is a review of some core mathematical concepts from the literature. It is primarily meant to consolidate notation, and to serve as a conceptual documentation reference for the rest of the package.

1. Background

1.1 Notation

For $K \in \mathbb{N}$ , let $[K] = \{0, 1, \dots, K - 1\}$ . For a finite set $S \subset \mathbb{N}$ of qubit indices, let $\mathcal{P}^S$ denote the set of unphased Pauli operators acting on those qubits. Note that we think of elements of $\mathcal{P}^S$ as functions mapping $S \rightarrow \{I, X, Y, Z\}$ , so that for any $P \in \mathcal{P}^S$ and $T \subseteq S$ , $P|_T$ denotes the restriction of $P$ to the qubit subset $T$ . Along these lines, for disjoint sets $S, T$ , and $P \in \mathcal{P}^S$ and $Q \in \mathcal{P}^T$ , $P \otimes Q$ denotes the element of $\mathcal{P}^{S \cup T}$ such that $P \otimes Q |_S = P$ and $P \otimes Q |_T = Q$ . This notation is helpful to avoid explicitly dealing with subsystem orderings, and to make it easy to describe restrictions.

For finite subsets $M \subset \mathbb{N}$ , we denote $\mathbb{Z}_2^M$ as the set of the bit strings whose elements are indexed by $M$ . Similarly to the above, we think of elements as functions mapping $M \rightarrow \mathbb{Z}_2$ , so that we may easily describe substrings in terms of restrictions of the index set.

Lastly, for a matrix $X$ , we use $| X \rangle\!\rangle$ to denote its vectorization. Based on the limited way in which we use this notation, it is not actually necessary to choose a specific vectorization convention. For a classical bit string $m$ , we use the shorthand $| m \rangle\!\rangle = | | m \rangle\langle m | \rangle\!\rangle$ .

1.2 Quantum instruments

A quantum operation producing classical bits $m \in \mathbb{Z}_2^M$ (for example, the result of measurement) is generally modeled as a linear map of the form:

\rho \mapsto \sum_{m \in \mathbb{Z}_2^M} \mathcal{E}_m(\rho) \otimes | m \rangle\langle m |,

where the set $\{\mathcal{E}_m: m \in \mathbb{Z}_2^M\}$ are completely positive, and $\sum_m \mathcal{E}_m$ is trace-preserving. A set of completely positive maps $\{\mathcal{E}_m\}$ satisfying these properties is called a quantum instrument. A unitary gate is a special case of an instrument with only one element that is a unitary operation.

2. Noisy Clifford-MCM-reset gates

The formalism utilized in this package assumes every gate in the gate set to be characterized consists of the following sequence of operations on $K$ qubits:

A Clifford operation on all qubits.
A mid-circuit projective measurement along $Z$ on some subset of qubits $M \subseteq [K]$ .
A mid-circuit reset to the $Z$ ground state on some subset of qubits $R \subseteq [K]$ .

This is a general class of operations that includes unitary Clifford gates, measurement, state preparation, and any combination of the above. Note that we assume measurement and reset are always along the $Z$ -axis for each qubit. While this is not strictly required, it is a common feature of many quantum computing modalities, and enables simplified representations and analysis.

As outlined in [1], [7] for the no-reset case, if a specific twirling strategy is applied to a noisy instance of such a gate, then the action of the resulting operation can be modeled mathematically as a uniform Pauli instrument. That is, within the quantum instrument notation, $\mathcal{E}_m = \mathcal{U}_mG$ , where $G$ is the Clifford unitary, and:

\mathcal{U}_m = \sum_{a,b \in \mathbb{Z}_2^M} \Lambda_{a,b} \otimes | m + a \rangle\!\rangle\langle\!\langle m + b |,

where each $\Lambda_{a,b}$ is a sub-normalized Pauli channel on the unmeasured qubits $N = [K] \setminus M$ . It is implied by this being a quantum instrument that $\sum_{a,b} \Lambda_{a,b}$ is also trace-preserving. Some notes:

The initial untwirled noise is modeled to include both “quantum” and “classical” errors: erroneous operations on the quantum registers, as well as mistakes in the measurement value reporting.
The noise map $\Lambda_{a,b}$ is independent of the measurement outcome $m$ .

In words, a single term in the above sum represents observing a measurement outcome of $m$ when the measured state was $| m + b \rangle$ (a misclassification if $b \neq 0$ ), and when the output state on the measurement register is $| m + a \rangle$ (the “wrong” state when $a \neq 0$ ). The map $\Lambda_{a,b}$ simultaneously encodes the action on the unmeasured qubits (conditioned on the measurement behavior) and the probability of the specific measurement behavior (through the normalization).

Note that we are not concerned here with the specifics of the twirling strategy: that such a strategy exists to put the channel into the above form is enough. Note that “finer” twirling strategies exist which can further restrict the form of the Pauli channels [2]; however, we take the above form as the most general mathematical representation under consideration.

In Lemma 1 of [7], it is shown that $\mathcal{E}_m = \mathcal{U}_mG$ can be rewritten as:

\mathcal{E}_m = \frac{1}{2^{2|M| + |N|}}\sum_{x, y \in \mathbb{Z}_2^M, Q \in \mathcal{P}^N} (-1)^{m \cdot (x + y)} \lambda^Q_{x, y}| Q \otimes Z^y \rangle\!\rangle\langle\!\langle G^\dagger(Q \otimes Z^x) |.

for some real numbers $\lambda_{x,y}^Q$ , which are called the fidelities of the instrument.

Adding reset to this picture is relatively straightforward. A noisy reset operation on qubits $R \subset [K]$ can be modeled according to the decomposition:

\frac{1}{2^{|R|}}\sum_{r \in \mathbb{Z}_2^{R}} \lambda_r | Z^r \rangle\!\rangle\langle\!\langle I_R |.

The above form explicitly utilizes the assumption that the reset is along the $Z$ -axis for each qubit. Noise in the operation is encoded in the reset fidelities $\lambda_r$ , which are simply indexed by qubit subsets.

An analog to Equation (1) that includes a reset operation at the end can be attained by simply composing it with Equation (2). This composition, after some simplification, yields:

\begin{split} \frac{1}{2^{2|M|+|N|}} \sum_{\substack{x \in \mathbb{Z}_2^M, y \in \mathbb{Z}_2^{M \cup R} \\ Q \in \mathcal{P}^{N\setminus R}}} (-1)^{m \cdot (x + y|_M)} \lambda^Q_{x,y} | Q \otimes Z^y \rangle\!\rangle\langle\!\langle G^\dagger(Q \otimes I_{N \cap R} \otimes Z^x) |. \end{split}

This decomposition indicates that the number of fidelities $\lambda^Q_{x, y}$ is

4^{|N \setminus R|} 2^{|M|}2^{|M \cup R|} = 4^{K}2^{|M|}2^{-|M \cup R|} = 4^K 2^{-|R \setminus M|}.

Consider limiting cases: (1) For no measurement or reset, $M = R = \emptyset$ , and the expression yields $4^K$ , the number of phase-less Pauli operators on $K$ qubits. (2) For $M = \emptyset$ and $R=[K]$ , this yields $2^K$ , which is the number of subsets of $[K]$ . Lastly, (3) for $M = [K]$ and $R = \emptyset$ , this yields $4^K$ . Note that this may initially seem surprising: for standard learning, measurement and reset are typically considered to have the same number of fidelities: $2^K$ . However, here the all-measurement gate in this context is understood not as a terminal measurement, but as an MCM, and as such fidelities corresponding to non-identity outputs of the measurement are also included (such as $y \neq 0$ in Equation (3)).

Finally, it is important to note that any twirling strategy yielding the form in Equation (1) in the non-reset case still applies equally well to the non-trivial reset case with a simple modification. The potential problem is that such a twirling strategy may require operations on measured qubits after the measurement. Therefore, if a qubit is measured and reset, implementing the strategy seemingly requires inserting operations within the gate between the measurement and reset. However, given the nature of reset, any such operations have no effect, and can simply be skipped while still yielding the same desired structure.

3. Path formalism

As described in the introduction, all learning algorithms for Pauli-type noise on Clifford gates are based around tracking how individual Paulis evolve through a circuit, picking up noise fidelity factors along the way. The expectation values of a circuit composed of such gates are therefore related simply to products of the underlying fidelities of the individual gates, and in this way model parameters can be inferred by inverting whatever sparse parameter-to-fidelity mapping is being assumed.

In [4], in the context of unitary gate sets, this is formalized into the Pattern Transfer Graph (PTG): a directed graph describing all possible experiments consisting of elements of the gate set and layers of single-qubit Clifford gates (assumed to be perfect, or “free”, operations). This was generalized in [7] to include gate sets with mid-circuit measurements. In both cases, each experiment is described by tracking a single Pauli operator through the circuit, under the assumption of a particular observable being computed at each measurement site. Here we do not directly review the graph or path constructions; however, we present the required facts for justifying them within our own notation and with resets included.

While the tracking of a single Pauli operator through the circuit may be intuitive in the unitary gate set case, it does not so obviously hold in the more general Clifford-MCM-reset case, due to the non-deterministic nature of measurement. The following proposition recovers this picture even in the more general case [7]:

Proposition 1

For any $x \in \mathbb{Z}_2^M$ , $y \in \mathbb{Z}_2^{M \cup R}$ , and $Q\in \mathcal{P}^{N \setminus R}$ , it holds that:

\langle\!\langle Q \otimes Z^y |\otimes \langle\!\langle Z^{x + y|_M} |_M \left(\sum_{m\in \mathbb{Z}_2^M} \mathcal{E}_m \otimes | m \rangle\!\rangle\right) = \lambda_{x,y}^Q\langle\!\langle G^\dagger(Q \otimes I_{N \cap R} \otimes Z^x) |

In other words, if we compute the expectation value $Z^{x + y|_M}$ on the classical register output by the MCM, then the “Heisenberg picture” evolution of an observable through the Clifford-MCM-reset gate maps $Q \otimes Z^y \rightarrow Q \otimes I_{N \cap R} \otimes Z^x$ . This fact alone enables recovery of PTG-like analysis beyond the unitary gate set case: with a particular post-processing of the measurement results, even gates with measurements and resets can be viewed as deterministically mapping one Pauli operator to another.

Before proving this, we write Equation (3) in a more suggestive way. Note that for $x, y \in \mathbb{Z}_2^M$ , it holds that $Z^{x+y} = \sum_{m \in \mathbb{Z}_2^M} (-1)^{m \cdot (x + y)}| m \rangle\langle m |$ , and as such (in super operator notation) we have:

\begin{split} \begin{aligned} \sum_{m \in \mathbb{Z}_2^M} \mathcal{E}_m \otimes | m \rangle\!\rangle_M = \frac{1}{2^{2|M|+|N|}}\sum_{\substack{x \in \mathbb{Z}_2^M, y \in \mathbb{Z}_2^{M \cup R} \\ Q \in \mathcal{P}^{N\setminus R}}} \lambda^Q_{x,y} | Q \otimes Z^y \rangle\!\rangle \langle\!\langle G^\dagger(Q \otimes I_{N \cap R} \otimes Z^x) | \otimes | Z^{x + y|_M} \rangle\!\rangle_M, \end{aligned} \end{split}

where we have applied Equation (3) and collected terms, and we use the subscript $M$ to indicate the classical measurement result register, which we treat above in a similar notation to the quantum registers.

Proof. It holds that:

\begin{split} \begin{aligned} &\langle\!\langle Q \otimes Z^y | \otimes \langle\!\langle Z^{x + y|_M} | \left(\sum_{m\in \mathbb{Z}_2^M} \mathcal{E}_m \otimes | m \rangle\!\rangle\right)\\ &= \langle\!\langle Q \otimes Z^y | \otimes \langle\!\langle Z^{x + y|_M} | \left( \frac{1}{2^{2|M|+|N|}} \sum_{\substack{a \in \mathbb{Z}_2^M, b \in \mathbb{Z}_2^{M \cup R} \\ P \in \mathcal{P}^{N\setminus R}}} \lambda^P_{a,b} | P \otimes Z^b \rangle\!\rangle\langle\!\langle G^\dagger(P \otimes I_{N \cap R} \otimes Z^a) | \otimes | Z^{a + b|_M} \rangle\!\rangle \right) \\ &=\frac{1}{2^{2|M| + |N|}}\sum_{\substack{a \in \mathbb{Z}_2^M, b \in \mathbb{Z}_2^{M \cup R} \\ P \in \mathcal{P}^{N \setminus R}}} \lambda^P_{a, b}\underbrace{\langle Q \otimes Z^y, P \otimes Z^b \rangle}_{2^{|M| + |N|} \delta_{P, Q}\delta_{y,b}} \underbrace{\langle Z^{x + y|_M}, Z^{a + b|_M} \rangle}_{2^{|M|}\delta_{x+y|_M,a+b|_M}}\langle\!\langle G^\dagger(P \otimes I_{N \cap R} \otimes Z^a) | \end{aligned} \end{split}

Lastly, observe that $\delta_{y,b}\delta_{x+y|_M,a+b|_M} = \delta_{y,b}\delta_{x,a}$ , and hence the final term above collapses down to the single term $\lambda^Q_{x,y} \opbra{G^\dagger(Q \otimes I_{N \cap R} \otimes Z^x)}$ , yielding the desired result.

See [7] for development beyond this point: the definition of the PTG, paths through the PTG, and the proof that any properly-defined path corresponds to an experiment.

References

[1] Stefanie J. Beale and Joel J. Wallman. Randomized compiling for subsystem measurements. 2023. URL: http://arxiv.org/abs/2304.06599, doi:10.48550/arXiv.2304.06599.

[2] Ewout van den Berg and Pawel Wocjan. Techniques for learning sparse Pauli-Lindblad noise models. Quantum, 8:1556, 2024. URL: https://quantum-journal.org/papers/q-2024-12-10-1556/, doi:10.22331/q-2024-12-10-1556.

[3] Edward H. Chen, Senrui Chen, Laurin E. Fischer, Andrew Eddins, Luke C. G. Govia, Brad Mitchell, Andre He, Youngseok Kim, Liang Jiang, and Alireza Seif. Disambiguating pauli noise in quantum computers. PRX Quantum, pages, 2026. doi:10.1103/69wc-gzl6.

[4] Senrui Chen, Zhihan Zhang, Liang Jiang, and Steven T. Flammia. Efficient Self-Consistent Learning of Gate Set Pauli Noise. PRX Quantum, 7(1):010305, 2026. URL: https://link.aps.org/doi/10.1103/1pnv-t9px, doi:10.1103/1pnv-t9px.

[5] Alexander Erhard, Joel J. Wallman, Lukas Postler, Michael Meth, Roman Stricker, Esteban A. Martinez, Philipp Schindler, Thomas Monz, Joseph Emerson, and Rainer Blatt. Characterizing large-scale quantum computers via cycle benchmarking. Nature Communications, 10(1):5347, 2019. doi:10.1038/s41467-019-13068-7.

[6] Steven T. Flammia. Averaged Circuit Eigenvalue Sampling. In 17th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2022), volume 232 of Leibniz International Proceedings in Informatics (LIPIcs), 4:1–4:10. Dagstuhl, Germany, 2022. Schloss Dagstuhl – Leibniz-Zentrum für Informatik. doi:10.4230/LIPIcs.TQC.2022.4.

[7] Zhihan Zhang, Senrui Chen, Yunchao Liu, and Liang Jiang. Generalized Cycle Benchmarking Algorithm for Characterizing Midcircuit Measurements. PRX Quantum, 6(1):010310, 2025. URL: https://link.aps.org/doi/10.1103/PRXQuantum.6.010310, doi:10.1103/PRXQuantum.6.010310.

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