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arXiv:2602.18375

Towards scalable multi-qubit optimal control via interaction decomposition in the diagonal frame

Bora Baran, Tommaso Calarco, Matthias M. Mueller, Felix Motzoi

quant-phSubmitted Feb 20, 2026• Announced Feb 23, 2026• 0 citations
v1 · Feb 20, 2026v2 · Jun 3, 2026
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Abstract

In this work, we introduce a general n-qubit formulation of control objectives that allows a control target to be specified in a diagonal frame, so that only the diagonal entries must be characterized, thus quadratically reducing the complexity of the cost functional in constrast to a full target matrix. We do so by representing any n-qubit unitary transformation as a diagonal phase map on the computational basis states, as they are naturally diagonalizable by unitarity. By using discrete derivative operators to analytically construct support-selective phase invariants, we enable to deterministically isolate and quantify any multi-qubit interactions encoded in the phase map. These phase invariants form a coordinate system for the formulation of specific control targets in terms of arbitrary desired multi-qubit interactions, without having to invert the diagonalization during the optimizatiion, solely relying on the experimentally accesible diagonal phases. To illustrate the framework, we synthesize two genuinely tripartite entangling gates, both, diagonal and non-diagonal. These are obtained with a single shaped microwave pulse, for a numerically simulated room-temperature nitrogen-vacancy center with a three qubit nuclear spin register, with durations of about a microsecond. These results represent a factor 10-100 reduction in operation time compared with the fastest existing NV-based entanglers that act on more than two qubits at once.

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18 pages, 6 Figures, 1 Table

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In this work, we introduce a general n-qubit formulation of control objectives that allows a control target to be specified in a diagonal frame, so that only the diagonal entries must be characterized, thus quadratically reducing the complexity of the cost functional in constrast to a full target matrix. We do so by representing any n-qubit unitary transformation as a diagonal phase map on the computational basis states, as they are naturally diagonalizable by unitarity. By using discrete derivative operators to analytically construct support-selective phase invariants, we enable to deterministically isolate and quantify any multi-qubit interactions encoded in the phase map. These phase invariants form a coordinate system for the formulation of specific control targets in terms of arbitrary desired multi-qubit interactions, without having to invert the diagonalization during the optimizatiion, solely relying on the experimentally accesible diagonal phases. To illustrate the framework, we synthesize two genuinely tripartite entangling gates, both, diagonal and non-diagonal. These are obtained with a single shaped microwave pulse, for a numerically simulated room-temperature nitrogen-vacancy center with a three qubit nuclear spin register, with durations of about a microsecond. These results represent a factor 10-100 reduction in operation time compared with the fastest existing NV-based entanglers that act on more than two qubits at once.
Quantum circuit synthesis describes the process of converting arbitrary unitary operations into a gate sequence of a fixed universal gate set, usually defined by the operations native to a given hardware platform. Most current synthesis algorithms are designed to synthesize towards a set of single qubit rotations and an additional entangling two qubit gate, such as CX, CZ, or the Molmer Sorensen gate. However, with the emergence of neutral atom based hardware and their native support for gates with more than two qubits, synthesis approaches tailored to these new gate sets become necessary. In this work, we present an approach to synthesize multi controlled phase gates using ZX calculus. By representing quantum circuits as graph like ZX diagrams, one can utilize the distinct graph structure of diagonal gates to identify multi controlled phase gates inherently present in some quantum circuits even if none were explicitly defined in the original circuit. We evaluate the approach on a wide range of benchmark circuits and compare them to the standard Qiskit synthesis regarding its circuit execution time for neutral atom based hardware with native support of multi controlled gates. Our results show possible advantages for current state of the art hardware and represent the first exact synthesis algorithm supporting arbitrary sized multi controlled phase gates.
A key task in quantum computation is the application of a sequence of gates implementing a specific unitary operation. However, the decomposition of an arbitrary unitary operation into simpler quantum gates is a nontrivial problem. Here we propose a general and robust protocol to decompose any target unitary into a sequence of Pauli rotations. The procedure involves identifying a commuting subset of Pauli operators having a high trace overlap with the target unitary, followed by a numerical optimization of their corresponding rotation angles. The protocol is demonstrated by decomposing several standard quantum operations. The applications of the protocol for quantum state preparation and quantum simulations are also described. Finally, we describe an NMR experiment implementing a three-body quantum simulation, wherein the above decomposition technique is used for the efficient realization of propagators.
Any unitary transformation of quantum computational networks is explicitly decomposed, in an exact and unified form, into a sequence of a limited number of one-qubit quantum gates and the two-qubit diagonal gates that have diagonal unitary representation in usual computational basis. This decomposition may be simplified greatly with the help of the properties of the finite-dimensional multiple-quantum operator algebra spaces of a quantum system and the specific properties of a given quantum algorithm. As elementary building blocks of quantum computation, the two-qubit diagonal gates and one-qubit gates may be constructed physically with one- and two-body interactions in a two-state quantum system and hence could be conveniently realized experimentally. The present work will be helpful for implementing generally any N-qubit quantum computation in those feasible two-state quantum systems and determining conveniently the time evolution of these systems in course of quantum computation.
A U(1)U(1)U(1) quantum rotor pairs a periodic angle with an integer-valued conjugate momentum, and occurs in molecular rotation, superconducting phase-charge circuits, and compact gauge fields. Coupling such a rotor coherently to qubits gives a hybrid register whose control structure is not inherited from either the oscillator-qubit or the qudit case. We develop a Clifford theory, together with a universal-control result, for registers of nnn qubits and rrr rotors. We classify all automorphisms of the hybrid phase space F22n×Zr×Tr\mathbb{F}_2^{2n}\times\mathbb{Z}^r\times\mathbb{T}^rF22n​×Zr×Tr that preserve the Weyl commutation relations, and give an explicit finite Clifford circuit for each one. The classification is directional: rotor momentum parity may control qubit Pauli operations within the Clifford group, while every nonzero qubit-controlled rotor momentum shift is non-Clifford. It also yields normal forms for the mixed qubit-rotor couplings and the exact minimum number of elementary mixed gates needed to synthesize them. Adding a rotor cosine potential and one fixed qubit-rotor conditional phase to the local Clifford operations gives universal control on the full Hilbert space in the strong operator topology. We then apply this structure in three settings: an exact controlled-shift realization of gauge-covariant matter hopping, which is necessarily non-Clifford; rotor phase estimation with direct angle readout and probe optimization under momentum-support and energy constraints; and finite Fourier transforms on rotor momentum codes, where the one-rotor transform for d=2sd=2^sd=2s compiles into O(s)O(s)O(s) momentum-selective and controlled-shift instructions and each cross-register Fourier factor is implemented by one quadratic rotor Clifford gate.
Global control provides a promising route to implementing multi-qubit gates without individual qubit addressing. This is especially appealing for permutation-invariant (PI) gates, whose symmetry is often broken when they are compiled into individually addressed one- and two-qubit gates. Important examples include SWAP, iSWAP\sqrt{i\text{SWAP}}iSWAP​, and the n-qubit controlled-Z gate, which is equivalent, up to two single-qubit Hadamard gates, to the multi-qubit Toffoli gate. Motivated by this global-control perspective, we show that all PI unitaries on an arbitrary number of qubits can be realized using the Tavis-Cummings (TC) interaction, the multi-qubit version of the Jaynes-Cummings interaction, together with global uniform z and x fields. Here, the nnn qubits are identically coupled to a single bosonic mode (oscillator), which is initialized in and returned to its vacuum state. A corollary is that all PI states, including GHZ and Dicke states, can be prepared using the same global control. For the case n=2 qubits, which is particularly important in quantum computing, we also find explicit pulse sequences for implementing all PI qubit unitaries that conserve angular momentum in the z direction, using only the TC interaction and global z fields. This includes controlled-Z, SWAP, and iSWAP\sqrt{i\text{SWAP}}iSWAP​.
Universal quantum computation requires the implementation of arbitrary control operations on the quantum register. In most cases, this is achieved by external control fields acting selectively on each qubit to drive single-qubit operations. In combination with a drift Hamiltonian containing interactions between the qubits, this allows the implementation of arbitrary gate operations. Here, we demonstrate an alternative scheme that does not require local control for all qubits: we implement one- and two-qubit gate operations on a set of target qubits indirectly, through a combination of gates on directly controlled actuator qubits with a drift Hamiltonian that couples actuator and target qubits. Experiments are performed on nuclear spins, using radio-frequency pulses as gate operations and magnetic-dipole couplings for the drift Hamiltonian.