Simulation of Two-Qubit Grover Algorithm in MBQC with Universal Blind Quantum Computation

TL;DR

This work tackles the privacy problem in cloud-based quantum computing by showing how MBQC, and its UBQC privacy protocol, can be efficiently simulated on circuit-based platforms. The authors develop a complete methodology to translate MBQC operations into circuit-native constructs, using gate teleportation, adaptive corrections, and measurement simulations within Qiskit, and validate it with a two-qubit Grover algorithm. They further integrate UBQC to demonstrate privacy-preserving computation, showing that function privacy can be maintained while performing a landmark quantum task on a circuit platform. The results indicate that MBQC-based privacy can be practically realized on current quantum hardware and simulators, paving the way for secure quantum cloud services with reduced qubit overhead and operational complexity.

Abstract

The advancement of quantum computing technology has led to the emergence of early-stage quantum cloud computing services. To fully realize the potential of quantum cloud computing, it is essential to develop techniques that ensure the privacy of both data and functions. Quantum computations often leverage superposition to evaluate a function on all possible inputs simultaneously, making function privacy a critical requirement. In 2009, Broadbent et al. introduced the Universal Blind Quantum Computation (UBQC) protocol, which is based on Measurement-Based Quantum Computation (MBQC) and provides a framework for ensuring both function and data privacy in quantum computing. Although theoretical results indicate an equivalence between MBQC and circuitbased quantum computation, translating MBQC into circuitbased implementations remains challenging due to higher qubit requirements and the complexity of the transformation process. Consequently, current quantum cloud computing platforms are limited in their ability to simulate MBQC efficiently. This paper presents an efficient method to simulate MBQC on circuit-based quantum computing platforms. We validate this approach by implementing the two-qubit Grover algorithm in the MBQC framework and further demonstrate blindness by applying the UBQC protocol. This work verifies the simulation of a blind quantum computation using the two-qubit Grover algorithm on a circuit-based quantum computing platform.

Figures (11)

• Figure 1: MBQC simulation based on Qiskit: $H$-gate. (a) The $H$-gate implementation in MBQC, where the qubit $q_0$ is the input and $q_1$ is the output; the arrow indicates the measurement angle in the $XY$ plane. (b) Test scenario: the input qubit $q_0$ is initialized in the $\ket{0}$ state. (c) After applying the $H$-gate, the state of $q_1$ collapses to either $\ket{0}$ or $\ket{1}$ with probability 0.5 upon measurement in the $Z$-basis.
• Figure 2: MBQC simulation based on Qiskit: $X$-gate. (a) The $X$-gate in MBQC, where the qubit $q_0$ is the input and $q_2$ is the output; the arrow indicates the measurement angle on the $XY$ plane. (b) Test scenario: the input qubit $q_0$ is initialized in the $\ket{0}$ state. (c) After applying the $X$-gate, the state of $q_2$ becomes $\ket{1}$ upon measurement in the $Z$-basis.
• Figure 3: MBQC simulation using Qiskit: $Z$-gate. (a) The $Z$-gate in MBQC, where qubit $q_0$ is the input and $q_2$ is the output; the arrow indicates the measurement angle in the $XY$ plane. (b) Test scenario: the input qubit $q_0$ is initialized in the $\ket{+}$ state. (c) After applying the $Z$-gate, the state of $q_2$ becomes $\ket{-}$, then collapses to $\ket{1}$ upon measurement in the $X$-basis.
• Figure 4: MBQC simulation using Qiskit: $T$-gate. (a) The $T$-gate in MBQC, where qubit $q_0$ is the input and $q_2$ is the output; the arrow indicates the measurement angle in the $XY$ plane. (b) Test scenario: the input qubit $q_0$ is initialized in the $\ket{+}$ state. (c) After applying the $T$-gate, the state of $q_2$ becomes $\ket{+_{\pi/4}} = \frac{1}{\sqrt{2}}(\ket{0} + e^{i\pi/4}\ket{1})$, which collapses to $\ket{0}$ with probability $\cos^2(\pi/8) \approx 0.8536$ upon $X$-basis measurement.
• Figure 5: MBQC simulation using Qiskit: $CZ$-gate. (a) The $CZ$-gate in MBQC, where qubits $q_0$ and $q_1$ are the inputs, and $q_4$ and $q_5$ are the outputs; the arrow indicates the measurement angle in the $XY$ plane. (b) Test scenario: the input state is $q_0, q_1 = \ket{1} \otimes \ket{+}$. After applying the $CZ$-gate, (c) the output state becomes $\ket{1} \otimes \ket{-}$. Both qubits then collapse to $\ket{1}$ upon $Z$-basis measurement on $q_4$ and $X$-basis measurement on $q_5$.