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arXiv:2507.09781

Decomposition of multi-qutrit gates generated by Weyl-Heisenberg strings

Daniele Trisciani, Marco Cattaneo, Zoltán Zimborás

Published Jul 13, 2025• 0 citations
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Abstract

Decomposing unitary operations into native gates is an essential step for implementing quantum algorithms. For qubit-based devices, where native gates are typically single- and two-qubit operations, a range of decomposition techniques have been developed. In particular, efficient algorithms exist for decomposing exponentials of Pauli strings while taking hardware topology in account. Motivated by the growing interest in qutrit-based quantum computing, we develop analogous decomposition methods for qutrit systems. Specifically, we introduce an algorithm that decomposes the exponential of an arbitrary tensor product of Weyl-Heisenberg operators (plus their Hermitian conjugation) into single- and two-qutrit gates. We further extend this approach to unitaries generated by Gell-Mann string (i.e., a tensor product of Gell-Mann matrices). Since both Gell-Mann matrices and Weyl-Heisenberg operators form (together with identity) complete operator bases of qutrit operators, we can use this result also to decompose any multi-qutrit gate that is diagonal up to single-qutrit rotations. As a practical application, we use our method to decompose the layers of the quantum approximate optimization algorithm for qutrit-based implementations of the graph k-coloring problem. For values of $k$ well-suited to qutrit architectures (e.g., $k=3$ or in general $k=3^n$), our approach yields significantly shallower circuits compared to qubit-based implementations, an advantage that grows with problem size, while also requiring a smaller total Hilbert space dimension. Finally, we also address the routing challenge in qutrit architectures that arises due to the limited connectivity of the devices. In particular, we generalize the Steiner-Gauss method, originally developed to reduce CNOT counts in qubit circuit, to optimize gate routing in qutrit-based systems.

DOI
10.48550/arxiv.2507.09781

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In this paper, we design quantum circuits for the exponential of scaled nnn-qubit Pauli strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity and XXX gates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with the nnn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Pauli-string is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1)(n+1)(n+1)-qubit systems can be constructed from nnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 222-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers and observe that the noisy simulation closely resembles the noiseless one when gate and idle errors are on the order of O(10−3)O(10^{-3})O(10−3) or smaller.
In this paper, we design quantum circuits for the exponential of scalednnn-qubit Pauli-strings using single-qubit rotation gates, Hadamard gate, and CNOT gates. A key result we derive is that any two Pauli-string operators composed of identity andXXXgates are permutation similar, and the corresponding permutation matrices are product of CNOT gates, with thennn-th qubit serving as the control qubit. Consequently, we demonstrate that the proposed circuit model for exponential of any Paulistring is implementable on low-connected quantum hardware and scalable i.e. quantum circuits for (n+1n+1n+1)-qubit systems can be constructed fromnnn-qubit circuits by adding additional quantum gates and the extra qubit. We then apply these circuit models to approximate unitary evolution for several classes of Hamiltonians using the Suzuki-Trotter approximation. These Hamiltonians include 2-sparse block-diagonal Hamiltonians, Ising Hamiltonians, and both time-independent and time-dependent Random Field Heisenberg Hamiltonians and Transverse Magnetic Random Quantum Ising Hamiltonians. Simulations for systems of up to 18 qubits show that the circuit approximation closely matches the exact evolution, with errors comparable to the numerical Trotterization error. Finally, we consider noise models in quantum circuit simulations to account for gate implementation errors in NISQ computers, and observe that the noisy numerical simulation closely resembles the noiseless one when gate and idle errors are on the order ofO(10−3)O\left(10^{-3}\right)O(10−3)or smaller.
We extend quantum circuit cutting to heterogeneous registers comprising mixed-dimensional qudits. By decomposing non-local interactions into tensor products of local generalised Gell-Mann matrices, we enable the simulation and execution of high-dimensional circuits on disconnected hardware fragments. We validate this framework on qubit--qutrit (222--333) interfaces, achieving exact state reconstruction with a Total Variation Distance of 0 within single-precision floating-point tolerance. Furthermore, we demonstrate the memory advantage in an 8-particle, dimension-8 system, reducing memory usage from 128 MB to 64 KB per circuit.
Although only two quantum states of a physical system are often used to encode quantum information in the form of qubits, many levels can in principle be used to obtain qudits and increase the information capacity of the system. To take advantage of the additional levels, a parameterization of unitary transformations in terms of experimentally realizable operations is needed. Many parameterizations of unitary 3 * 3 matrices (U(3)) exist. One decomposition of a general unitary matrix can be expressed as the product of an exponential of a diagonal matrix and an exponential of an off-diagonal matrix. This decomposition is relevant for controlling superconducting qutrits using fixed-frequency resonant control pulses. This decomposition is numerically confirmed to allow the parameterization of any element in U(3). It is shown that a simple setting of parameter ranges of parameters can easily lead to an over-parameterization, in the sense that several different sets of values for the parameters produce the same element in U(3). This fact is demonstrated using the Walsh-Hadamard (WH) matrix as an example, which is also a special qutrit gate of practical interest. The different decompositions are shown to be related, and the relationships between them are presented using general methods. The shortest path needed for the implementation of a qutrit gate is found. Other parameterizations obtained by other analytic means, which can be advantageous for various reasons, are also discussed.
The problem of finding efficient decompositions of multiqubit gates is of importance for quantum computing, especially, in application to existing noisy intermediate-scale quantum devices, whose resources are substantially limited. Here we propose a decomposition scheme for a generalized NNN-qubit Toffoli gate with the use of $2N\ensuremath{-}3$ two-qutrit gates for arbitrary connectivity. The fixed number of the required additional levels (the choice of qutrits is optimal) and the use of the iSWAP gate as a native operation make our approach directly applicable for ongoing experiments with superconducting quantum processors. Specifically, we present a blueprint of the realization of the proposed scheme for the Aspen-9 processor supporting quantum operations with qutrits.
Quantum circuits of a general quantum gate acting on multiple ddd-level quantum systems play a prominent role in multi-valued quantum computation. We first propose a new recursive Cartan decomposition of semi-simple unitary Lie group U(3n)U(3^n)U(3n) (arbitrary nnn-qutrit gate). Note that the decomposition completely decomposes an n-qutrit gate into local and non-local operations. We design an explicit quantum circuit for implementing arbitrary two-qutrit gates, and the cost of our construction is 21 generalized controlled X (GCX) and controlled increment (CINC) gates less than the earlier best result of 26 GGXs. Moreover, we extend the program to the nnn-qutrit system, and the quantum circuit of generic nnn-qutrit gates contained 4196⋅32n−4⋅3n−1−(n22+n4−2932)\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})9641​⋅32n−4⋅3n−1−(2n2​+4n​−3229​) GGXs and CINCs is presented. Such asymptotically optimal structure is the best known result so far.