Blind quantum computing with different qudit resource state architectures

TL;DR

This work extends blind quantum computing from qubits to qudits by developing a finite-field and integer-ring framework for qudits and showing that qudit versions of brickwork, open-ended cluster, and decorated cluster resource states enable server-blind execution. It introduces approximately universal single-qudit gate sets built from diagonal hidden rotations and identifies appropriate generalized non-Clifford diagonal gates (T_d^F) for prime-power dimensions, with continuous-angle schemes yielding exact universality. It analyzes hiding strategies for two-qudit entangling gates across various qudit resource architectures (brickwork, open-ended, decorated) and generalizes graph-hiding techniques to qudit graphs, including overhead considerations. The results lay the groundwork for secure, high-dimensional cloud-based quantum computation and point to future directions such as multi-client configurations and integration into fault-tolerant schemes.

Abstract

We discuss how blind quantum computing generalizes to multi-level quantum systems (qudits), which offers advantages compared to the qubit approach. Here, a quantum computing task is delegated to an untrusted server while simultaneously preventing the server from retrieving information about the computation it performs, the input, and the output, enabling secure cloud-based quantum computing. In the standard approach with qubits, measurement-based quantum computing is used: single-qubit measurements on cluster or brickwork states implement the computation, while random rotations of the resource qubits hide the computation from the server. We generalize finite-sized approximately universal gate sets to prime-power-dimensional qudits and show that qudit versions of the cluster and brickwork states enable a similar server-blind execution of quantum algorithms. Furthermore, we compare the overheads of different resource state architectures and discuss which hiding strategies apply to alternative qudit resource states beyond graph states.

Figures (5)

• Figure 1: Measurement-based quantum computing on the qudit cluster state. $(a)$ Single-qudit gates are implemented via local measurements on one-dimensional resource state chains, each processing a logical qudit $\ket{\psi}$. The information flows from left to right, and the state $\ket{\psi}$ is subject to $H D_{\Vec{\phi_k}}$, $k \in \{1,\hdots, 5\}$, in each step. $(b)$ The upper and lower horizontal chains correspond to logical qudits being processed and $Z$ and $Y$ measurements on the physical qudits in between place entangling gates at desired positions while simultaneously deleting the measured qudits from the resource state.
• Figure 2: Blind quantum computing with the qubit brickwork state Broadbent_2009. On the elementary brickwork state unit, two logical qubits are processed, one along each horizontal resource state chain. The identity and any of the gates within $\{ H, S, T, CX \}$ can be realized via local measurements in the $X$ basis, rotated by angles in the finite-sized set $\mathcal{A}$ of Eq. \ref{['eq:random-angles-qubits']}. $(a)$ Measurement pattern to implement the Hadamard gate. The single-qubit gates indicate the chosen measurement bases, where every Hadamard gate corresponds to an $X$ measurement while $HS$ corresponds to a measurement in the $X$ basis, rotated by $S^\dagger$. $(b)$ Implementation of the controlled-$X$ gate on an elementary qubit brickwork state unit. $(c)$ Arrangement of elementary units (highlighted) into the qubit brickwork state. Whenever logical qubits are not part of any elementary unit, they experience the identity, so transport without processing, if the respective physical qubits are measured in $X$ due to $H^2 = I_2$.
• Figure 3: To hide the positions of entangling gates from the server, different strategies, corresponding to different resource state architectures, can be chosen beyond the qubit brickwork state in Fig. \ref{['fig:brickwork-patterns']}. $(a)$ The open-ended cluster state allows for universal quantum computing without unhidden and, thus, structure-revealing $Z$ deletion measurements clusteruniversality-XYmeasurements. Here, the highlighted unit corresponds to $C_2$ in Eq. \ref{['eq:layer-operator']} if the qubits in the first column are measured in $X$. $(b)$ Hair implantation technique, where each cluster state qubit is decorated with a two-qubit chain (light grey) that allows for simulating both rotated $X$ and $Z$ basis measurements without performing $Z$ measurements. This naturally generalizes to graph states, including cluster states in higher dimensions. $(c)$ Graph hiding, where ancillary qubits (dark violet) are initialized in the $X$ or $Z$ basis (white text), which cannot be distinguished by the server, depending on where entangling gates should be placed. The controlled-phase $CZ$ gate then does not have any effect on the $Z$ basis qubits (greyed out edges) while $X$ basis qubits are entangled with the resource. Similar structures are referred to as the square brickwork state in Ref. ImprovedBrickworkState.
• Figure 4: Qudit brickwork state elementary unit, where the dashed edge corresponds to $CZ^{-1}$. The elementary units can be arranged into the qudit brickwork state in the same manner as for qubits, Fig. \ref{['fig:brickwork-patterns']}$(c)$. Each elementary unit supports the realization of any diagonal gate, the Hadamard gate, or the $CX$ entangling gate on the two logical qudits.
• Figure 5: Elementary unit of lattice size $4 \cross 6$ for the open-ended qudit cluster state and $n=4$ logical qudits. The numbers indicate the flow of information, the logical qudits, from the input on the left towards the output on the right if $X$ measurements are performed on all qudits except the output due to the qudit mirror being performed. Measurements in the first column serve to implement diagonal gates on the input. Instead, measurements in the fifth column allow the measurement-based implementation of diagonal gates, conjugated by the Hadamard gate, on the output qudits. Rotated $X$ bases measurements in the first row and a column between the first and fifth allow us to realize entangling gates between different pairs of output qudits, chosen by the column, as indicated in the figure.