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arXiv:2607.20895

Heralded high-dimensional module-based quantum computation

Xiao Zhang, Wen-Qiang Liu, Hai-Rui Wei

quant-phSubmitted Jul 23, 2026• Announced Jul 24, 2026• 0 citations
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Abstract

Parity measurements have been explored as building blocks for preparing and discriminating entangled states, as well as for implementing quantum computation. We first develop two alternative high-dimensional generalized parity modules, and then propose a procedure for constructing high-dimensional generalized module-based controlled-NOT gate. The construction of module-based quantum computing introduced here is deterministic, heralded, insensitive to the dimensionality of the computing basis, and postselection technique is not required. The result shows that out of $(d-1)!$ generalized parity modules, only modules $ \mathcal{P}=(j\ominus i)\bmod d$ and $ \mathcal{P}=(j \oplus i)\bmod d$ can be used as building blocks for high-dimensional quantum computing. Furthermore, we proposed an optical nondestructive scheme for implementing generalized parity module through quantum nondemolition measurements, and the success of the parity module is heralded by photon-number-resolving detectors and single-photon detectors.

Journal reference
Optics Express 34 (15): 28401-28415 (2026)
Comments
14 pages, 4 Figures

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Parity measurements have been explored as building blocks for preparing and discriminating entangled states, as well as for implementing quantum computation. We first develop two alternative high-dimensional generalized parity modules, and then propose a procedure for constructing high-dimensional generalized module-based controlled-NOT gate. The construction of module-based quantum computing introduced here is deterministic, heralded, insensitive to the dimensionality of the computing basis, and postselection technique is not required. The result shows that out of (d−1)!(d-1)!(d−1)! generalized parity modules, only modules P=(j⊖i) mod d \mathcal{P}=(j\ominus i)\bmod dP=(j⊖i)modd and P=(j⊕i) mod d \mathcal{P}=(j \oplus i)\bmod dP=(j⊕i)modd can be used as building blocks for high-dimensional quantum computing. Furthermore, we proposed an optical nondestructive scheme for implementing generalized parity module through quantum nondemolition measurements, and the success of the parity module is heralded by photon-number-resolving detectors and single-photon detectors.
Parity measurements have been explored as building blocks for preparing and discriminating entangled states, as well as for implementing quantum computation. We first develop two alternative high-dimensional generalized parity modules, and then propose a procedure for constructing a high-dimensional generalized module-based controlled-NOT gate. The construction of module-based quantum computing introduced here is deterministic, heralded, and insensitive to the dimensionality of the computing basis, and a postselection technique is not required. The result shows that out of ( d − 1)! generalized parity modules, only modules P =( j ⊖ i )mod d and P =( j ⊕ i )mod d can be used as building blocks for high-dimensional quantum computing. Furthermore, we proposed an optical nondestructive scheme for implementing a generalized parity module through quantum nondemolition measurements, and the success of the parity module is heralded by photon-number-resolving detectors and single-photon detectors.
Measurements play an important role in quantum computing (QC), either by providing the nonlinearity required for two-qubit gates (linear optics QC), or by implementing a quantum algorithm using single-qubit measurements on a highly entangled initial state (cluster state QC). Parity measurements can be used as building blocks for preparing arbitrary stabilizer states, and, together with one-qubit gates, are universal for quantum computing. Here we generalize parity gates by using a higher-dimensional (qudit) ancilla. This enables us to go beyond the stabilizer or graph state formalism and prepare other types of multiparticle entangled states. The generalized parity module introduced here can prepare in one shot, heralded by the outcome of the ancilla, a large class of entangled states, including Greenberger-Home-Zeilinger states GHZn{\mathrm{GHZ}}_{n}GHZn​, Wn{\mathrm{W}}_{n}Wn​, Dicke states Dn,k{\mathcal{D}}_{n,k}Dn,k​, and, more generally, certain sums of Dicke states, like Gn{G}_{n}Gn​ states used in secret sharing. For Wn{\mathrm{W}}_{n}Wn​ states it provides an exponential gain compared to linear-optics-based methods.
Quantum gates are the essential block for quantum computer. High-dimensional quantum gates exhibit remarkable advantages over their two-dimensional counterparts for some quantum information processing tasks. Here we present a family of entanglement-based optical controlled-SWAP gates on C2⊗Cd⊗Cd\mathbb{C}^{2}\otimes \mathbb{C}^{d}\otimes \mathbb{C}^{d}C2⊗Cd⊗Cd. With the hybrid encoding, we encode the control qubits and target qudits in photonic polarization and spatial degrees of freedom, respectively. The circuit is constructed using only (2+3d)(2+3d)(2+3d) (d≥2d\geq 2d≥2) linear optics, beating an earlier result of 14 linear optics with d=2d=2d=2. The circuit depth 5 is much lower than an earlier result of 11 with d=2d=2d=2. Besides, the fidelity of the presented circuit can reach 99.4\%, and it is higher than the previous counterpart with d=2d=2d=2. Our scheme are constructed in a deterministic way without any borrowed ancillary photons or measurement-induced nonlinearities. Moreover, our approach allows d>2d>2d>2.
Compared with traditional two-dimensional qubits, higher-dimensional qubits offer a larger state space for storing and processing information, which can manifest stronger non-classical correlations, simplify circuit complexity, reduce computational resources, and enhance loss tolerance in photon-based quantum information processing. To harness the potential, essential research on preparing qudit-based high-dimensional entangled states is necessary. In the paper, we propose the preparation procedure of arbitrary four-dimensional two-qudit entanglement and three-qudit entanglement for photon systems implemented with error-heralded quantum devices and controlled-not gates. Moreover, as the practical errors that grow out of defective photon scattering, i.e., coupling strength restriction of the quantum-dot-cavity system and cavity leakage imperfection, can be turned into the response of the single-photon detectors, the generation of arbitrary high-dimensional entanglement appears in an error-heralded way, and their fidelities are robust, enhancing experimental accessibility.
Based on two element operations, called controlled-path gate and entangler, the non-destructive multi-photon parity check could be deterministically realized. With these operations, the realization of loss-tolerant optical quantum computation, including the generation of parity states, the realization of single-particle operation and two-particle CNOT operation, could be very simple but deterministic and highly efficient. Moreover, the character of detected free of single photons in the computation, except of the parity-check, will make the requirement of single-photon resources much relax. Compared with the former linear optical schemes, this scheme with cross-phase modulation is simpler but deterministic, and more efficient but with less resource, which will make it more feasible and suitable for large-scale quantum computation.
Abstract Quantum gates are the essential block for quantum computers. High‐dimensional quantum gates exhibit remarkable advantages over their 2D counterparts for some quantum information processing tasks. Here, a family of entanglement‐based optical controlled‐SWAP gates on is presented. With the hybrid encoding, the control qubits and target qudits are encoded in photonic polarization and spatial degrees of freedom, respectively. The circuit is constructed using only () linear optics, beating an earlier result of 14 linear optics with . The circuit depth five is much lower than an earlier result of 11 with . Besides, the fidelity of the presented circuit can reach 99.4%, and it is higher than the previous counterpart with . The scheme is constructed in a deterministic way without any borrowed ancillary photons or measurement‐induced nonlinearities. Moreover, the approach allows .