The Parity Flow Formalism: Tracking Quantum Information Throughout Computation

TL;DR

The paper addresses the challenge of understanding information flow in quantum circuits by introducing the Parity Flow formalism, which attaches parity labels to track how Pauli content evolves under Clifford gates and how non-Clifford actions arise in the encoded space. The authors develop covariant (pushforward) and contravariant (pullback) tracking, and further unify them with a combined flow tableau to enable simultaneous updates and efficient inversions, with explicit rules and runtime considerations. They demonstrate how the framework supports encoding changes and parallelization via stabilizer codes and auxiliary qubits, reducing circuit depth and revealing new parallelizable operations. The approach provides a scalable, interpretable, and hardware-aware tool for circuit synthesis, debugging, and optimization, with linear-time label updates per Clifford gate and manageable memory overhead.

Abstract

We propose the Parity Flow formalism, a method for tracking the information flow in quantum circuits. This method adds labels to quantum circuit diagrams such that the action of Clifford gates can be understood as a recoding of quantum information. The action of non-Clifford gates in the encoded space can be directly deduced from those labels without backtracking. An application of flow tracking is to design resource-efficient quantum circuits by changing any present encoding via a simple set of rules. Finally, the Parity Flow formalism can be used in combination with stabilizer codes to further reduce quantum circuit depth and to reveal additional operations that can be implemented in parallel.

Figures (10)

• Figure 1: (a) Labeled quantum circuit with tracked single qubit Clifford gates (in white) and $\mathrm{CNOT}$ gates, and untracked interleaved Pauli rotations defined as $R_P(\alpha)=\exp(iP\alpha/2)$ (grey gates). The labels allow us to read off the logical effect of the physically applied rotations (see the grey boxes). The circuit demonstrates the logical application of the Heisenberg model pengetal2022heisenberg$e^{i\alpha\bar{Y}_{1}\bar{Y}_{2}}e^{i\beta\bar{X}_{1}\bar{X}_{2}}e^{i\gamma\bar{Z}_{1}\bar{Z}_{2}}$ as $C_{3}e^{-i\alpha X_{2}}C_{2}e^{i\beta X_{2}}e^{i\gamma Z_{1}}C_{1}$ [cf. Eq. \ref{['eq:alternating_unitary']}]. Note that the label transformation of the $S^\dagger$ gate can be obtained from that of three sequential $S$ gates. (b) Update rules for general labels $\ell_{X\small{j}}$ and $\ell_{Z\small{j}}$ under the action of a set of generators of the Clifford group, which are defined via their corresponding operators [see Eq. \ref{['eq:update_2']}].
• Figure 2: (a) $\mathrm{CNOT}$ circuit to obtain the effect of $\bar{Z}_{1}\bar{Z}_{2}$ on an initially empty auxiliary qubit $a$. The Clifford phases are omitted, since the action of the $\mathrm{CNOT}$ gates is trivial. (b) Encoding circuit to implement all logical operators of the Heisenberg model $e^{i\alpha\bar{Y}_{1}\bar{Y}_{2}}e^{i\beta\bar{X}_{1}\bar{X}_{2}}e^{i\gamma\bar{Z}_{1}\bar{Z}_{2}}$ in parallel after encoding redundantly with an auxiliary qubit. Dots and crosses on top of an index denote stabilized and destabilized operators, respectively. After the encoding circuit, a physical $Y$ rotation on the auxiliary qubit results in a valid logical $-\bar{Y}_1\bar{Y}_2$ operator, since $\textcolor{xlog}{\overset{$\:\bullet$}{a}}$ can be omitted from the label $\ell_{{X}_{a}}$ for interpreting its logical effect. The Clifford phases $\langle i^{\eta_{\langle 1\rangle}}\dotsc\rangle$$i^{\eta_1}\dotsc$ are tracked on top of the circuit according to Eq. \ref{['eq:S_X_clifford_phase']}. (c) The resulting encoding of the circuit in (b) showing the stabilizer at the time step of the rotations. The circles show the three qubits, labeled with the non-stabilizer violating logical operators (i.e., no crossed indices) corresponding to the local physical operators on the corners of the triangle. The triangle indicates that the stabilizer $\bar{X}_{a}$ is mapped to $X_{1}Z_{2}Y_{a}$ by covariant tracking, revealing that the logical operator on $Y_{a}$ must be equal to the logical operator that corresponds to operator $X_{1}Z_{2}$.
• Figure A.1: The group operation $\circledast$ written in the flow label notation a) in the example of Eq. \ref{['exp:groupop']}, b) in the general case. For ${\color{xlog}\bm{\xi_{}}}$ and ${\color{zlog}\bm{\zeta_{}}}$ just take the symmetric difference $\triangle$ of the sets. For the phase multiply the phases with the sign correction, which is $(-1)$ to the power of the number of elements $\vert{\color{zlog}\bm{\zeta_{1}}}\cap{\color{xlog}\bm{\xi_{2}}}\vert$ in the intersection marked by the rounded rectangles (identifying $\textcolor{xlog}{\langlej\rangle}$ and $\textcolor{zlog}{j}$), corresponding to the number of anticommuting swaps in order to produce the standard form of Pauli operators.
• Figure B.2: The action of a logical Pauli operator $\bar{P}$ on the logical state $\vert \bar{\psi}\rangle$ gets pushed forward to the action $C_\ast\bar{P}$ on the physical state $\vert \psi\rangle$ after the Clifford circuit $C$. $C$ acts on $\bar{P}$ by left conjugation.
• Figure B.3: The action of a physical Pauli operator $P$ on the physical state $\vert\psi\rangle$ gets pulled back to the action $C^\ast P$ on the logical state $\vert\bar{\psi}\rangle$ before the Clifford circuit $C$. $C$ acts on $P$ by right conjugation.

Theorems & Definitions (106)

• Definition A.1
• Remark A.2
• Proposition A.3
• proof
• Proposition A.4
• proof
• Remark A.5
• Remark A.6
• Remark A.7
• Remark A.8