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Quantum 2.0 Conference and Exhibition

Gauge Transformations for Measurement-Based Quantum Compiling

Sebastiano Corli, Enrico Prati

Published Jan 1, 2025• 2 citations
Publisher

Abstract

We introduce a direct method for measurement-based quantum compiling, converting any unitary circuit to a class of graph states. A set of graphical rules and a system of equations implement the process.
DOI
10.1364/quantum.2025.qw4b.5

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The measurement-based architecture is a paradigm of quantum computing, relying on the entanglement of a cluster of qubits and the measurements of a subset of it, conditioning the state of the unmeasured output qubits. While methods to map the gate model circuits into the measurement-based are already available via intermediate steps, we introduce a paradigm for quantum compiling directly converting any quantum circuit to a class of graph states, independently from its size. This method relies on the stabilizer formalism to describe the register of the input qubits. An equivalence class between graph states able to implement the same circuit is defined, giving rise to a gauge freedom when compiling in the MBQC frame. The graph state can be rebuilt from the circuit and the input by employing a set of graphical rules. A system of equations describes the overall process. Compared with measurement calculus, the ancillary qubits are reduced by 50% on the quantum Fourier transform and 75% on the Quantum Approximate Optimization Algorithm, still preserving similar scaling laws for the number of entangling gates.
Stabilizer states, along with Clifford manipulations (unitary transformations and measurements) thereof—despite being efficiently simulable on a classical computer—are an important tool in quantum information processing, with applications to quantum computing, error correction, and networking. Graph states, defined on a graph, are a special class of stabilizer states that are central to measurement-based quantum computing, all-photonic quantum repeaters, distributed quantum computing, and entanglement distribution in a network. All stabilizer states are local-Clifford equivalent to graph states. In this paper, we review the stabilizer framework and extend it by incorporating general stabilizer measurements such as multi-qubit joint projections. We provide an explicit procedure—using Karnaugh maps from Boolean algebra—for converting arbitrary stabilizer gates into tableau operations of the CHP formalism for efficient stabilizer manipulations. We derive graphical rules for arbitrary stabilizer manipulations of graph states, including multi-qubit stabilizer projections and unitaries. We implement the graphical rulebook resulting from above into a MATLAB simulator with a GUI. A user of this tool, e.g., for research in quantum networks, will not require any background in quantum information or the stabilizer framework.
Abstract Quantum computers may offer multiple advantages, consisting of quantum speed-up when performing hard computation tasks, better compression and expressivity, reduced power consumption, and the capability of simulating natively elementary processes by representing their constituents by qubit. Therefore, chemical, nuclear and elementary particle reactions or Hamiltonian dynamics have been simulated consistently with the number of available qubits. The compact use of resources is of paramount importance for both simulating larger problems or, in prospect, introduce more robust quantum error correction routines to extend the depth of the circuits. Measurement-based quantum computing (MBQC) consists of a virtualization of gate model quantum computing over a limited subset of gate operation involving intermediate measurement processes as well, particularly suitable for photonic hardware. Here, we review the method recently introduced for direct compiling gate model circuit based on unitary processes over the measurement-based scheme. Surprisingly, in addition to better performances with respect to previously known methods, we find a gauge freedom which makes possible to even improve the efficiency. Such gauge invariance provides graphical rules, as happens with Feynman rules in perturbative quantum mechanics. Compared to Measurement Calculus, the ancillary qubits are reduced by 50% on QFT and 75% on QAOA algorithms.