[Paper Review] Multi-party Quantum Computation

Adam Smith|ArXiv.org|Nov 6, 2001

Quantum Information and CryptographyComputer Science46 references17 citations

TL;DR

This paper presents a secure multi-party quantum computation (MPQC) protocol that tolerates up to $ n/6 $ dishonest parties by introducing a verifiable quantum secret sharing (VQSS) scheme capable of handling up to $ t < n/4 $ cheaters. The key contribution is proving that VQSS with $ t < n/4 $ is optimal and using it to construct a secure MPQC protocol under the same threshold, leveraging quantum error-correcting codes and idealized simulation models in a quantum information-theoretic framework.

ABSTRACT

We investigate definitions of and protocols for multi-party quantum computing in the scenario where the secret data are quantum systems. We work in the quantum information-theoretic model, where no assumptions are made on the computational power of the adversary. For the slightly weaker task of verifiable quantum secret sharing, we give a protocol which tolerates any t &lt; n/4 cheating parties (out of n). This is shown to be optimal. We use this new tool to establish that any multi-party quantum computation can be securely performed as long as the number of dishonest players is less than n/6.

Motivation & Objective

• To formalize secure multi-party quantum computation (MPQC) in the quantum information-theoretic model, where adversaries have unbounded computational power.
• To define and construct a verifiable quantum secret sharing (VQSS) protocol that ensures both completeness and soundness under adversarial control of up to $ t < n/4 $ parties.
• To establish the optimal threshold for VQSS and use it as a foundational primitive to enable secure MPQC when $ t < n/6 $.
• To analyze the security of MPQC protocols through ideal vs. real model simulation, ensuring that cheating parties learn no more than allowed by the function output and their inputs.

Proposed method

• Proposes a two-level quantum sharing protocol using 2-good trees to enable verifiable distribution of quantum secrets among $ n $ parties.
• Employs quantum error-correcting codes and subspace projection techniques to ensure that corrupted parties cannot alter the shared state without detection.
• Uses the concept of neighborhood and trace states ($ ST_B({/cal C}) $) to model adversarial corruption in quantum codes, particularly for CSS codes.
• Applies the ideal functionality model to define security: a protocol is secure if it is simulatable by an ideal process that guarantees correctness and privacy.
• Introduces a simulation-based proof technique to show that the VQSS protocol is indistinguishable from an ideal secret sharing functionality, even under $ t < n/4 $ corruption.
• Constructs a top-level sharing protocol based on VQSS to enable distributed quantum computation, where each party holds part of the input and output state.

Experimental results

Research questions

• RQ1 What is the maximum number of dishonest parties that can be tolerated in a verifiable quantum secret sharing (VQSS) protocol?
• RQ2 Can a secure multi-party quantum computation (MPQC) protocol be constructed using VQSS as a primitive, and what is its threshold of resilience?
• RQ3 Is the bound $ t < n/4 $ for VQSS optimal, and can it be improved beyond this limit?
• RQ4 How do quantum error-correcting codes and subspace structures relate to the security of distributed quantum protocols?
• RQ5 Can the security of MPQC be proven using simulation-based definitions in the quantum setting, even when adversaries have quantum capabilities?

Key findings

• The VQSS protocol is shown to tolerate up to $ t < n/4 $ dishonest parties, and this bound is proven to be optimal.
• A secure MPQC protocol is constructed that tolerates up to $ t < n/6 $ dishonest parties, using VQSS as a foundational primitive.
• The paper proves that $ t < n/4 $ is the information-theoretic upper bound for VQSS, meaning no protocol can achieve higher resilience under the same assumptions.
• The set of states that can arise from adversarial corruption is characterized as $ ST_B({/cal C}) $, which captures the idealized effect of entangled errors on quantum codes.
• For CSS codes, the set $ {/cal C}_B $, which includes all possible states reachable via local operations on the code, strictly contains $ ST_B({/cal C}) $, showing that not all states are adversarially realizable.
• The paper establishes that $ N_B^{pure}({/cal C}) = ST_B^{pure}({/cal C}) $ for pure states, but this equality fails for mixed states, highlighting a key distinction in adversarial modeling.