Quantum delegation with an off-the-shelf device

TL;DR

A new model is put forth where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a single round, providing the first relativistic (one-round), two-prover zero-knowledge proof system for QMA.

Abstract

Given that reliable cloud quantum computers are becoming closer to reality, the concept of delegation of quantum computations and its verifiability is of central interest. Many models have been proposed, each with specific strengths and weaknesses. Here, we put forth a new model where the client trusts only its classical processing, makes no computational assumptions, and interacts with a quantum server in a single round. In addition, during a set-up phase, the client specifies the size $n$ of the computation and receives an untrusted, off-the-shelf (OTS) quantum device that is used to report the outcome of a single measurement. We show how to delegate polynomial-time quantum computations in the OTS model. This also yields an interactive proof system for all of QMA, which, furthermore, we show can be accomplished in statistical zero-knowledge. This provides the first relativistic (one-round), two-prover zero-knowledge proof system for QMA. As a proof approach, we provide a new self-test for n EPR pairs using only constant-sized Pauli measurements, and show how it provides a new avenue for the use of simulatable codes for local Hamiltonian verification. Along the way, we also provide an enhanced version of a well-known stability result due to Gowers and Hatami and show how it completes a common argument used in self-testing.

Figures (6)

• Figure 1: During the set-up the verifier selects an off-the-shelf device based on the required size of the problem instance. Afterward, the verifier is free to select any language and instance and interacts in a single round with both the prover and the off-the-shelf device, leading to the accept/reject output of the verifier.
• Figure 2: Off-the-shelf (OTS) proof system. $P$ is the quantum prover, $V$ is the classical verifier, and $D$ a rudimentary off-the-shelf-device; each arrow represents a single classical message. We can interchangeably think of the model as a strengthening of the verifier in $\AM$ ($V$ having access to an additional OTS device $D$), or a weakening of ${\MIP}^*$ (one of the provers, $D$, is severely restricted).
• Figure 3: Low-weight linearity test.
• Figure 4: Low-weight anti-commutation test.
• Figure 5: $x$ is an instance in $L \in \QMA$. $V_x$ is a $poly(\lvert\mspace{1mu} x \mspace{1mu}\rvert)$-size quantum circuit. $H_x$ is a $poly(\lvert\mspace{1mu} x \mspace{1mu}\rvert)$-qubit 6-local Hamiltonian of $XZ$-type. $\widehat{\mathcal{G}}_x$ is a nonlocal game with $poly(\lvert\mspace{1mu} x \mspace{1mu}\rvert)$-bit questions and $poly(\lvert\mspace{1mu} x \mspace{1mu}\rvert)$-bit answers.

Theorems & Definitions (60)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4: Informal version of \ref{['thm:simulation of history states']}
• Theorem 1.5: Informal version of \ref{['rigidity']}
• Theorem 1.6: Informal version of \ref{['thm:GH']}
• Definition 2.1: Polynomial-time uniform circuit family
• Definition 2.2: Q-CIRCUIT
• Definition 2.3
• Lemma 2.4