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arXiv:2607.00210

Classification and Exact Local Masking in Finite-Field Clifford Dual-Unitary Circuits

Basanta R Pahari

quant-phSubmitted Jun 30, 2026• Announced Jul 2, 2026• 0 citations
v1 · Jun 30, 2026v2 · Jul 4, 2026
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Abstract

We classify two-qudit finite-field Clifford dual-unitary gates and apply the classification to exact local masking and operator transport in homogeneous brickwork circuits. Under ordered one-qudit Clifford equivalence, the dual-unitary locus contains $q-2$ perfect-tensor cores, one rank-one core, and one SWAP core. The perfect cores are indexed by \[ δ=\det B=\det C, \] and are related to the ordered cross-ratio $λ$ of the associated $[4,2,3]_q$ MDS configuration through \[ λ=\fracδ{δ-1}. \] Homogeneous repetition separates these cores into five transport phases whose algebraic codimensions in $\operatorname{Sp}(4,q)$ are $0,1,2,3,4$. The one-site Weyl edge channels determine exact local-masking distances. Perfect-tensor circuits attain \[ d_1(t)=4t, \qquad d_2(t)=4t-2, \] whereas delayed erasers satisfy \[ d_1(t)=4t-2, \qquad d_2(t)=4t-4 \] for $t\geq 2$. Consequently, sufficiently short quantum messages are completely hidden from every one- or two-qudit output subsystem, even when the input is entangled with a reference, while remaining exactly recoverable from the full output. For $q=3$, we construct an explicit perfect-tensor Clifford gate from two inverse SUM gates. Exhaustive Weyl-support searches for $t=1,2,3$ reproduce the predicted masking distances, and a one-period Choi-channel calculation for a four-qutrit periodic circuit gives numerical residual leakage below $2\times 10^{-16}$. Under the coherent perturbation considered here, local leakage is linear in the perturbation strength, whereas the infidelity of recovery using the ideal inverse is quadratic near the perfect point.

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The one-site Weyl edge channels determine exact local-masking distances. Perfect-tensor circuits attain [ d_1(t)=4t, \qquad d_2(t)=4t-2, ] whereas delayed erasers satisfy [ d_1(t)=4t-2, \qquad d_2(t)=4t-4 ] for t≥2t\geq 2t≥2. Consequently, sufficiently short quantum messages are completely hidden from every one- or two-qudit output subsystem, even when the input is entangled with a reference, while remaining exactly recoverable from the full output.
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Dual-unitary quantum circuits have recently attracted attention as an analytically tractable model of many-body quantum dynamics. Consisting of a 1+1D lattice of 2-qudit gates arranged in a 'brickwork' pattern, these models are defined by the constraint that each gate must remain unitary under swapping the roles of space and time. This dual-unitarity restricts the dynamics of local operators in these circuits: the support of any such operator must grow at the effective speed of light of the system, along one or both of the edges of a causal light cone set by the geometry of the circuit. Using this property, it is shown here that for 1+1D dual-unitary circuits the set of width- w conserved densities (constructed from operators supported over w consecutive sites) is in one-to-one correspondence with the set of width- w solitons – operators which, up to a multiplicative phase, are simply spatially translated at the effective speed of light by the dual-unitary dynamics. A number of ways to construct these many-body solitons (explicitly in the case where the local Hilbert space dimension d=2 ) are then demonstrated: firstly, via a simple construction involving products of smaller, constituent solitons; and secondly, via a construction which cannot be understood as simply in terms of products of smaller solitons, but which does have a neat interpretation in terms of products of fermions under a Jordan-Wigner transformation. This provides partial progress towards a characterisation of the microscopic structure of complex many-body solitons (in dual-unitary circuits on qubits), whilst also establishing a link between fermionic models and dual-unitary circuits, advancing our understanding of what kinds of physics can be explored in this framework.
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.
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We establish that transversal diagonal gates on exact qubit codes of distance at least two can be exhaustively classified within a finite search space. Although code-preserving physical gates can have continuous phases, any projective logical action induced by a transversal diagonal gate can always be realized by a physical gate of finite order. Crucially, we prove that this physical order has an upper bound determined solely by the code parameters. This order bound enables the subset-sum-linear-programming (SSLP) framework introduced in arXiv:2504.20847 to exhaustively enumerate all candidate discrete symmetries, after which the Knill-Laflamme equations determine the remaining continuous code parameters. Applying this method to the nontrivial diagonal sector of exact ((6,2,3))((6,2,3))((6,2,3)) codes, SSLP yields 666666 candidate symmetry classes, of which exactly 151515 are realizable, supporting full projective diagonal logical cyclic groups of orders 222, 333, and 555. In particular, for order 555, the code is unique up to local unitaries and qubit permutations, equivalent to the code constructed in arXiv:2504.20847. This code also attains the minimum length n=6n=6n=6 among exact ((n,2,≥3))((n, 2, \ge 3))((n,2,≥3)) codes whose full projective transversal group cannot be conjugated into the projective Clifford group.