Parallel remote state preparation for fully device-independent verifiable blind quantum computation

TL;DR

This work addresses secure delegation of quantum computation under device-independent assumptions by combining parallel self-testing with the Fitzsimons–Kashefi verifiable blind quantum computation protocol. It develops a parallel self-testing framework that certifies $n$ Bell pairs and Pauli measurements in the $XY$ plane and computational basis, while handling complex conjugation and using a logarithmic-size client-question set. The approach leverages parallel CHSH and a triple CHSH inequality to construct a local isometry that acts independently of the preparation bases, enabling composability with FK VBQC and maintaining blindness and verifiability. The protocol realizes fully device-independent VBQC with practical features such as single-qubit client measurements, polynomial classical processing, and a polynomially bounded question set, offering robust scalability within the device-independent paradigm and paving the way for more fault-tolerant implementations. Overall, the paper presents a rigorous, scalable framework for device-independent quantum computation delegation that integrates robust parallel state certification with well-established VBQC verification mechanisms, potentially impacting cloud-based quantum computing with reduced trust assumptions and stronger security guarantees.

Abstract

We introduce a device-independent two-prover scheme in which a classical verifier can use a simple untrusted quantum measurement device (the client device) to securely delegate a quantum computation to an untrusted quantum server. To do this, we construct a parallel self-testing protocol to perform device-independent remote state preparation of $n$ qubits and compose this with the unconditionally secure universal verifiable blind quantum computation (VBQC) scheme of Fitzsimons and Kashefi [Phys. Rev. A 96, 012303 (2017)]. Our self-test achieves a multitude of desirable properties for the application we consider, giving rise to practical and fully device-independent VBQC. It certifies parallel measurements of all cardinal and intercardinal directions in the $XY$ plane as well as the computational basis, uses few input questions (of size logarithmic in $n$ for the client and a constant number communicated to the server), and requires only single-qubit measurements to be performed by the client device.

Figures (4)

• Figure 1: A modified partial swap isometry followed by phase kickback unitaries, acting on the state $\lvert\psi\rangle_{\mathcal{A} \mathcal{B}}$, which is used to self-test the state $\lvert\Phi^{+}\rangle$ and measurements of $\{\sigma_{\mathrm{x}}, \sigma_{\mathrm{y}}, \sigma_{\mathrm{z}}\}$ or $\{\sigma_{\mathrm{x}}, \sigma_{\mathrm{y}}, - \sigma_{\mathrm{z}}\}$, given a maximal violation of the triple CHSH inequality. The unitary operators on Alice's subsystem are simply $X_{\mathcal{A}} = A_{1}$, $Y_{\mathcal{A}} = - A_{2}$, and $Z_{\mathcal{A}} = A_{3}$. The unitary operators $\hat{X}_{\mathcal{B}}$, $\hat{Y}_{\mathcal{B}}$, and $\hat{Z}_{\mathcal{B}}$ are regularized versions of $X_{\mathcal{B}}$, $Y_{\mathcal{B}}$, and $Z_{\mathcal{B}}$ respectively, which are each defined in terms of the operators $D_{j,k}$ and $E_{j,k}$.
• Figure 2: The circuit describing the action of the local isometry $V = K W$ on the state $\lvert\psi\rangle_{\mathcal{A} \mathcal{B}}$. The isometries $W_{\mathcal{A}} \colon \mathcal{A} \to \mathcal{A} \otimes \mathcal{A}^{\prime}$ and $W_{\mathcal{B}} \colon \mathcal{B} \to \mathcal{B} \otimes \mathcal{B}^{\prime}$ and the local isometry $W = W_{\mathcal{A}} \otimes W_{\mathcal{B}}$ are defined by the first "swap" stage of the circuit (preceding the dotted line), in which a maximally entangled state is extracted. In the second "phase kickback" stage (succeeding the dotted line), denoted by $K = K_{\mathcal{A}} \otimes K_{\mathcal{B}}$ for isometries $K_{\mathcal{A}} \colon \mathcal{A} \to \mathcal{A} \otimes \mathcal{A}^{\prime\prime}$ and $K_{\mathcal{B}} \colon \mathcal{B} \to \mathcal{B} \otimes \mathcal{B}^{\prime\prime}$, possible complex conjugation in the presence of a Pauli $\sigma_{\mathrm{z}}$ operator is accounted for.
• Figure 3: The circuit describing the action of the local isometry $V^{(j)} = K^{(j)} W^{(j)}$ on the state $\lvert\psi\rangle_{\mathcal{A} \mathcal{B}}$. In the first "swap" stage (preceding the dotted line), denoted by $W^{(j)} = W_{\mathcal{A}}^{(j)} \otimes W_{\mathcal{B}}^{(j)}$, a maximally entangled state is extracted. In the second "phase kickback" stage (succeeding the dotted line), denoted by $K^{(j)} = K_{\mathcal{A}}^{(j)} \otimes K_{\mathcal{A}}^{(j)}$, possible complex conjugation in the presence of a Pauli $\sigma_{\mathrm{z}}$ operator is accounted for. The full isometry $V = V^{(n)} \dots V^{(1)}$ is a parallel version of this circuit. It is defined by applying the circuit for each $j$ successively, appending ancillae states in $\mathcal{A}_{j}^{\prime}$ and $\mathcal{B}_{j}^{\prime}$ for each swap stage, and $\mathcal{A}_{j}^{\prime\prime}$ and $\mathcal{B}_{j}^{\prime\prime}$ for each phase kickback stage. Each isometry $V^{(k)}$ is defined to act trivially on all ancilla spaces with $j < k$.
• Figure 4: A protocol that performs the blind preparation of $n$ qubits in appropriate bases on the server-side subsystem, followed by VBQC.

Theorems & Definitions (65)

• Theorem 1: Informal \ref{['thm:state_preparation']}
• Definition 2: Schatten $p$-norm
• Lemma 3
• proof
• Definition 4: Self-testing of states and real measurements
• Definition 5: Self-testing of states and complex measurements
• Lemma 6
• proof
• Lemma 7
• proof