[Paper Review] Quantum k-means algorithm based on Trusted server in Quantum Cloud Computing

Changqing Gong, Dong Zhao-yang+2 more|arXiv (Cornell University)|Nov 9, 2020

Quantum Computing Algorithms and Architecture17 references4 citations

TL;DR

This paper proposes a trusted-server-assisted quantum k-means algorithm in quantum cloud computing that leverages quantum homomorphic encryption (QHE) to outsource computationally intensive subroutines—specifically $\text{SwapTest}$ and $\text{GroverOptim}$—to the cloud. By using a trusted server to manage complex $T$-gate key updates, the client reduces its computational burden, achieving a time complexity reduction from $O(Mnkt)$ to $O(M\log(n)\sqrt{k}t)$, while maintaining security via quantum one-time pad (QOTP).

ABSTRACT

We propose a quantum k-means algorithm based on quantum cloud computing that effectively solves the problem that the client can not afford to execute the same quantum subroutine repeatedly in the face of large training samples. In the quantum k-means algorithm, the core subroutine is the Quantum minimization algorithm (GroverOptim), the client needs to repeat several Grover searches to find the minimum value in each iteration to find a new clustering center, so we use quantum homomorphic encryption scheme (QHE) to encrypt the data and upload it to the cloud for computing. After calculation, the server returns the calculation result to the client. The client uses the key to decrypt to get the plaintext result. It reduces the computing pressure for the client to repeat the same operation. In addition, when executing in the cloud, the key update of T-gate in the server is inevitable and complex. Therefore, this paper also proposes a T-gate update scheme based on trusted server in quantum ciphertext environment. In this scheme, the server is divided into trusted server and semi-trusted server. The semi-trusted server completes the calculation operation, and when the T-gate is executed in the circuit, the trusted server assists the semi-trusted server to calculate the T-gate, and then randomly generates a key and uploads it to the semi-trusted server. The trusted server assists the client to complete the key update operation, which once again reduces the pressure on the client and improves the efficiency of the quantum homomorphic encryption scheme. And on the basis of this scheme, the experiment is given by using IBM Qiskit to give the subroutine of quantum k-means. The experimental results show that the scheme can realize the corresponding computing function on the premise of ensuring security.

Motivation & Objective

• Address the high computational burden on clients when repeatedly executing $\text{GroverOptim}$ in quantum k-means with large training datasets.
• Enable secure outsourcing of quantum machine learning subroutines to the cloud while preserving data confidentiality.
• Reduce client-side complexity in managing $T$-gate key updates during quantum homomorphic encryption operations.
• Design a hybrid trusted/semi-trusted server model to improve efficiency and security in quantum ciphertext computation.
• Validate the feasibility and correctness of the proposed scheme through quantum circuit simulations using IBM Qiskit.

Proposed method

• Encrypt quantum state data using a quantum homomorphic encryption (QHE) scheme based on quantum one-time pad (QOTP) for secure cloud computation.
• Outsource the execution of $\text{SwapTest}$ and $\text{GroverOptim}$ subroutines to a semi-trusted quantum cloud server.
• Introduce a trusted server to assist in $T$-gate key updates, simplifying the client’s role to only key management and decryption.
• Decompose the quantum circuit into trusted and semi-trusted components, where the trusted server handles $T$-gate operations and key updates.
• Implement and simulate the full quantum k-means pipeline using IBM Qiskit, including similarity testing and minimum value retrieval in ciphertext.
• Apply amplitude amplification and repeated execution to mitigate decoherence-induced errors in noisy quantum environments.

Experimental results

Research questions

• RQ1 How can the client securely outsource the repeated execution of $\text{GroverOptim}$ in quantum k-means without compromising data privacy?
• RQ2 What is the role of a trusted server in simplifying $T$-gate key update operations within a quantum homomorphic encryption framework?
• RQ3 To what extent does the proposed scheme reduce client-side computational load compared to native execution?
• RQ4 How do noise and decoherence in real quantum devices affect the accuracy of outsourced quantum subroutines like $\text{GroverOptim}$ and $\text{SwapTest}$?
• RQ5 Can the proposed QHE-based trusted-server model be generalized to other quantum machine learning algorithms with complex, repeated operations?

Key findings

• The proposed scheme reduces the time complexity of quantum k-means from $O(Mnkt)$ to $O(M\log(n)\sqrt{k}t)$, significantly improving scalability.
• The use of a trusted server for $T$-gate key updates reduces client-side complexity and improves the efficiency of quantum homomorphic encryption.
• Experimental results using IBM Qiskit confirm the correct functionality of $\text{SwapTest}$ and $\text{GroverOptim}$ in the encrypted domain, with expected quantum states emerging with high probability.
• Noise and decoherence in real quantum devices cause state degradation and bit flips, leading to reduced accuracy and probability distribution shifts in output states.
• Amplitude amplification and multiple executions help mitigate error effects, showing that the scheme remains robust under noise with proper repetition.
• The scheme is secure under the QOTP-based QHE model and is applicable to other delegated quantum computing tasks requiring complex, repeated operations.