Reducing depth and measurement weights in Pauli-based computation

TL;DR

Pauli-based computation (PBC) leverages adaptive sequences of commuting Pauli measurements on separable magic-state qubits, but its resource costs scale with the T-gate count $t$. The authors introduce a pre-compilation step that uses a 1WQC pre-structure to derive nontrivial upper bounds on Pauli weights and computational depth, and they further reduce weights with a greedy classical algorithm, achieving up to about 30% average reductions in Pauli weight for Clifford-dominated circuits with moderate $t$, and substantial CNOT-count reductions relative to ZX-calculus peers for certain circuit classes. They also demonstrate a depth reduction to match the 1WQC depth $d_{1W}$ and a weight–depth trade-off, plus a universal incPBC variant that uses only constant-weight measurements at the cost of incompatibility and more measurements. Together, these results significantly improve the practicality of PBC as a circuit-compiler primitive and broaden MBQC’s toolbox for efficient, fault-tolerant quantum computation, including dedicated comparisons to state-of-the-art compilers showing substantial improvements in the relevant regimes.

Abstract

Pauli-based computation (PBC) is a universal measurement-based quantum computation model steered by an adaptive sequence of independent and compatible Pauli measurements on separable magic-state qubits. Here, we propose several new techniques for reducing the weight of the Pauli measurements and their associated \textsc{cnot} complexity; we also demonstrate how to decrease this model's computational depth. We start by proving new upper bounds on the required weights and computational depth, obtained via a pre-compilation step. We also propose a heuristic algorithm that can contribute reductions of over 30\% to the average weight of Pauli measurements (and associated \textsc{cnot} count) when simulating and compiling Clifford-dominated random quantum circuits with up to 22 $T$ gates and over 20\% for instances with larger $T$ counts. This PBC-compilation scheme, boosted by the heuristic algorithm, outperforms state-of-the-art compilers for the former circuits, reducing the \textsc{cnot} count by 18\% to 96\% compared with the values achieved by other techniques. In contrast, for the latter circuits with larger $T$ counts, it leads to a number of \textsc{cnot}s roughly 30\% larger. Finally, inspired by known state-transfer methods, we introduce incPBC, a universal model for quantum computation requiring a larger number of (now incompatible) Pauli measurements of weight at most 2.

Figures (18)

• Figure 1: The usual workflow to perform a Pauli-based computation (PBC). Starting from a Clifford+$T$ quantum circuit with $n$ qubits, $t$$T$ gates, and $w$ computational-basis readout measurements, step ① consists of transforming the circuit using magic-state injection. The result of such a transformation is an adaptive Clifford circuit on $n+t$ qubits and with $t+w$ measurements. Such a circuit can be easily transformed into a PBC: The classical computer efficiently finds the next Pauli measurement, decides its outcome (classically) when possible, and queries the quantum computer when necessary. The work in Sec. \ref{['sec: Improvements I - connection to 1WC']} targets step ②: A classically efficient pre-compilation step is added before making the PBC translation. This is leveraged to establish new upper bounds for the weight of the Pauli measurements and the computational depth of PBC. The work in Sec. \ref{['sec: Greedy algorithm']} addresses step ③: A new heuristic (classical) algorithm replaces $P_j$ with an equivalent measurement $P_j^{\prime}$ with lower weight, before the quantum hardware is queried.
• Figure 2: Fault-tolerant implementation of the $T$ gate via the well-known $T$-gadget, using only stabilizer operations and classical feedforward.
• Figure 3: Implementation of state transfer via single- and two-qubit Pauli measurements, up to a Pauli correction, $P$, that depends on all three measurement outcomes. Grey boxes with rounded edges are used throughout to represent projective measurements; the outcome of each of these measurements is stored in memory and accessible for future use (if needed).
• Figure 4: Implementation of the cnot gate using state transfer; the gate is applied to the (arbitrary) two-qubit input state $\ket{\psi}$ up to a two-qubit Pauli operator $P$ which depends on the outcomes of the measurements. Note that the second measurement involves only the first and last qubits, as it has an identity on the second qubit, that is, it consists of the measurement $Z_1X_3$ (having only weight 2).
• Figure 5: Implementation of (a) the Hadamard gate and (b) the $T$ gate using state transfer. Note that the unitary transformations are applied to the (arbitrary) input state $\ket{\psi}$ up to a single-qubit Pauli operator $P$ which depends on the measurement outcomes.

Theorems & Definitions (13)

• Remark 1: Removing output qubits
• Remark 2: Universality of PBC
• Theorem 1: Improved weights
• Corollary 2: Average weight upper bound
• Theorem 3: Improved depth
• Theorem 4: Weight-depth trade-off
• Lemma 5: Single-layer computation
• proof
• Corollary 6
• proof