Verifiable cloud-based variational quantum algorithms

TL;DR

This work tackles verifiability and channel-loss resilience in cloud-based variational quantum algorithms (VQAs) by extending Shingu et al.'s ancilla-driven quantum computation approach. It introduces trap-qubit–based verification, encrypted measurements, and Bell-pair sharing to achieve verifiability with minimal quantum resources for the client and a modest resource overhead for the server. Theoretical guarantees show input, output, and algorithm blindness, along with a bound on the probability of undetected server deviation, and the protocol is shown to offer improved tolerance to channel loss compared to prior schemes. The approach aims to make cloud-based VQAs more practical for clients with limited quantum capabilities while maintaining security and correctness, potentially enabling real-world deployment of verifiable quantum cloud services.

Abstract

Variational quantum algorithms (VQAs) have shown potential for quantum advantage with noisy intermediate-scale quantum (NISQ) devices for quantum machine learning (QML). However, given the high cost and limited availability of quantum resources, delegating VQAs via cloud networks is a more practical solution for clients with limited quantum capabilities. Recently, Shingu et al.[Physical Review A, 105, 022603 (2022)] proposed a variational secure cloud quantum computing protocol, utilizing ancilla-driven quantum computation (ADQC) for cloud-based VQAs with minimal quantum resource consumption. However, their protocol lacks verifiability, which exposes it to potential malicious behaviors by the server. Additionally, channel loss requires frequent re-delegation as the size of the delegated variational circuit grows, complicating verification due to increased circuit complexity. This paper introduces a new protocol to address these challenges and enhance both verifiability and tolerance to channel loss in cloud-based VQAs.

Figures (5)

• Figure 1: Circuit for the $J(\phi)$ operator: The prepared qubits are $\ket{\psi}_R$ and $\ket{+}_A$, where the subscripts $R$ and $A$ denote the register qubits and ancillary qubit, respectively. The highlighted section represents the measurement of the ancillary qubit $\ket{+}_A$ in the basis $\left\{\frac{1}{\sqrt{2}}(\ket{0} \pm exp(-i\phi)\ket{1})\right\}$. After measurement, the operation $X^{s}HR_Z(\phi)$ is obtained, where $X$ is the Pauli $X$ operator and $s$ is the measurement result.
• Figure 2: General variational quantum circuit: All qubits are initialized in the state $\ket{0}$. The operator $En$ encodes classical data $\vec{x}$ into the quantum state $En(\vec{x})\ket{0}^{\otimes w}$. The unitary operator $U(\vec{\theta}) = \prod_{i=1}^{n} U_i(\vec{\theta}^i)$ represents the variational layers, forming a specifically designed ansatz, where each $U_i(\vec{\theta^i})$ corresponds to the $i$-th layer of the $n$ variational layers.
• Figure 3: Circuit for the two-qubit controlled gates: The server interacts with the client to obtain 6 $J(\phi)$ operators with parameters $\alpha$, $\beta$, $\gamma$, and $\delta$.
• Figure 4: (a) The universal gate patterns: Composed of multiple blocks, each representing a gate pattern. (b) The gate pattern circuit: Consisting of 8 $J(\phi)$ operators and 2 $CZ$ gates. (c) Realization of the $J(\phi)$ operator: The server sends one half of a Bell pair to the client, who measures in the basis $M\left(-\frac{k\pi}{4}\right)$ with result $s_0$. The server then performs operations on the ancillary and register qubits, including a fixed coupling operation $(H_R \otimes I_A)CX_{RA}$, where $CX$ denotes the controlled-X gate and $I$ is the identity gate, followed by measurements in the $Z$ basis or in the $M(\phi')$ basis on the ancillary qubits with results $s_1$ to $s_3$.
• Figure 5: Brief process of the proposed protocol: The server sends an ancillary qubit, as one half of a Bell pair, to the client, who measures it in the basis $M(-\frac{k\pi}{4})$. The client then sends the reception status back to the server, requesting a resend if the qubit is lost. The server performs encrypted measurements as instructed by the client and returns the results, enabling the client to verify the server's honesty and calculate encrypted measurement angles and gradients for the optimizer.

Theorems & Definitions (6)

• Theorem 1
• proof
• Theorem 2
• proof
• Theorem 3
• proof