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arXiv:2411.12485 · Physical Review Applied

Measurement-based quantum compiling via gauge invariance

Sebastiano Corli, Enrico Prati

Published Mar 24, 2026• 2 citations
arXivHTMLPDF

Abstract

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.

DOI
10.1103/lpkt-zgfn

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•Mar 1, 2024•0 citations
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.
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.
Abstract The measurement-based quantum computing paradigm relies on entangling the qubits of a register into a graph state and on measuring subsets of such register in order to condition the unmeasured qubits. The computation therefore, instead of being carried on only by one-qubit and two-qubits logic gates, is largely based on entanglement and measurement processes. Compared to the gate model architecture, in MBQC the overhead in terms of computation resources to synthesize logical qubits is higher. While gate model relies on a register of logical qubits whose number is set at the beginning of the computation and remains constant during the computation, in the MBQC the input qubits outnumber the output ones, due to the destructive nature of measurement processes. Still, we analytically prove and experimentally confirm that MBQC can be efficiently emulated on a classical software, reaching equal loads with respect to the gate-based approach in terms of average runtime and storage of memory. The numerical results confirm that despite the potential computational overhead due to the high entanglement of the initial graph state, the MBQC paradigm carries similar computational complexity, both in terms of time and memory, with respect to the gate-based approach.
The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions, a so-called standard form, which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus enabling a significant decrease in circuit depth for certain applications. We show that groups of fully commuting operators can be implemented using fully-parallel, i.e., non-adaptive, measurements within our protocol. In addition, we discuss how such circuits can be implemented in constant quantum depths by employing quantum teleportation. Finally, we demonstrate the utility of our technique on two examples of high practical relevance: the Quantum Approximate Optimization Algorithm (QAOA) and the Variational Quantum Eigensolver (VQE).
Measurement-based quantum computing (MBQC), an alternate paradigm for formulating quantum algorithms, can lead to potentially more flexible and efficient implementations as well as to theoretical insights on the role of entanglement in a quantum algorithm. Using the graph-theoretical ZX-calculus, we describe and apply a general scheme for reformulating quantum circuits as MBQC implementations. After illustrating the method using the two-qubit Deutsch–Jozsa algorithm, we derive a ZX graph-diagram that encodes a general MBQC implementation for the three-qubit Deutsch–Jozsa algorithm. This graph describes an 11-qubit cluster state on which single-qubit measurements are used to execute the algorithm. Particular sets of choices of the axes for the measurements can be used to implement any realization of the oracle. In addition, we derive an equivalent lattice cluster state for the algorithm.
The paradigm of measurement-based quantum computing (MBQC) starts from a highly entangled resource state on which unitary operations are executed through adaptive measurements and corrections ensuring determinism. This is set in contrast to the more common quantum circuit model, in which unitary operations are directly implemented through quantum gates prior to final measurements. In this work, we incorporate concepts from MBQC into the circuit model to create a hybrid simulation technique, permitting us to split any quantum circuit into a classically efficiently simulatable Clifford-part and a second part consisting of a stabilizer state and local (adaptive) measurement instructions—a so-called standard for —which is executed on a quantum computer. We further process the stabilizer state with the graph state formalism, thus, enabling a significant decrease in circuit depth for certain applications. We show that groups of mutually-commuting operators can be implemented using fully-parallel, i.e. non-adaptive, measurements within our protocol. In addition, we discuss how groups of mutually commuting observables can be simulatenously measured by adjusting the resource state, rather than performing a costly basis transformation prior to the measurement as it is done in the circuit model. Finally, we demonstrate the utility of our technique on two examples of high practical relevance—the Quantum Approximate Optimization Algorithm and the Variational Quantum Eigensolver (VQE) for the ground-state energy estimation of the water molecule. For the VQE, we find a reduction of the depth by a factor of 4 to 5 using measurement patterns vs. the standard circuit model. At the same time, since we incorporate the simultaneous measurements, our patterns allow us to save shots by a factor of at least 3.5 compared to measuring Pauli strings individually in the circuit model.