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arXiv:2004.10486 · Physical Review A

Secure multiparty quantum computation with few qubits

Victoria Lipińska, Jérémy Ribeiro, Stephanie Wehner

Published Aug 7, 2020• 16 citations
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Abstract

We consider the task of secure multiparty distributed quantum computation on a quantum network. We propose a protocol based on quantum error correction which reduces the number of necessary qubits. That is, each of the $n$ nodes in our protocol requires an operational workspace of ${n}^{2}+4n$ qubits, as opposed to the previously shown $\mathrm{\ensuremath{\Omega}}\left(({n}^{3}+{n}^{2}{s}^{2})logn\right)$ qubits, where $s$ is a security parameter. Additionally, we reduce the communication complexity by a factor of $O({n}^{3}log(n))$ qubits per node compared to existing protocols. To achieve universal computation, we develop a distributed procedure for verifying magic states, which allows us to apply distributed gate teleportation and which may be of independent interest. We showcase our protocol in a small example for a seven-node network.

DOI
10.1103/physreva.102.022405

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We consider the task of secure multiparty distributed quantum computation on a quantum network. We propose a protocol based on quantum error correction which reduces the number of necessary qubits. That is, each of the nnn nodes in our protocol requires an operational workspace of n2+4n{n}^{2}+4nn2+4n qubits, as opposed to the previously shown \ensuremathΩ((n3+n2s2)logn)\mathrm{\ensuremath{\Omega}}\left(({n}^{3}+{n}^{2}{s}^{2})logn\right)\ensuremathΩ((n3+n2s2)logn) qubits, where sss is a security parameter. Additionally, we reduce the communication complexity by a factor of O(n3log(n))O({n}^{3}log(n))O(n3log(n)) qubits per node compared to existing protocols. To achieve universal computation, we develop a distributed procedure for verifying magic states, which allows us to apply distributed gate teleportation and which may be of independent interest. We showcase our protocol in a small example for a seven-node network.
Secure multiparty quantum computation (MPQC) protocol is a cryptographic primitive allowing error-free distributed quantum computation to a group of nnn mutually distrustful quantum nodes even when some quantum nodes disobey the instructions of the protocol. Here we suggest a modified MPQC protocol that adopts unconventional quantum error-correcting codes and as a consequence reduces the number of qubits required for the protocol execution. In particular, the replacement of the self-dual Calderbank-Shor-Steane quantum error-correcting codes with triply even ones permits us to avoid the previously indispensable but resource-intensive procedure of the ``magic-state'' verification. Besides, since every extra qubit reduces the credibility of physical devices, our suggestion makes the MPQC protocol more accessible for the near-future technology by reducing the number of necessary qubits per quantum node from ${n}^{2}+\mathrm{\ensuremath{\Theta}}(r)n$, where rrr is the security parameter, to n2+3n{n}^{2}+3nn2+3n.
Secret sharing and multiparty computation (also called "secure function evaluation") are fundamental primitives in modern cryptography, allowing a group of mutually distrustful players to perform correct, distributed computations under the sole assumption that some number of them will follow the protocol honestly. This paper investigates how much trust is necessary -- that is, how many players must remain honest -- in order for distributed quantum computations to be possible. We present a verifiable quantum secret sharing (VQSS) protocol, and a general secure multiparty quantum computation (MPQC) protocol, which can tolerate any \left[ {\frac{{n - 1}} {2}} \right] cheaters among n players. Previous protocols for these tasks tolerated \left[ {\frac{{n - 1}} {4}} \right] and \left[ {\frac{{n - 1}} {6}} \right] cheaters, respectively. The threshold we achieve is tight -- even in the classical case, "fair" multiparty computation is not possible if any set of n/2 players can cheat. Our protocols rely on approximate quantum errorcorrecting codes, which can tolerate a larger fraction of errors than traditional, exact codes. We introduce new families of authentication schemes and approximate codes tailored to the needs of our protocols, as well as new state purification techniques along the lines of those used in faulttolerant quantum circuits.
Secret sharing and multiparty computation (also called “secure function evaluation”) are fundamental primitives in modern cryptography, allowing a group of mutually distrustful players to perform correct, distributed computations under the sole assumption that some number of them will follow the protocol honestly. This paper investigates how much trust is necessary – that is, how many players must remain honest – in order for distributed quantum computations to be possible. We present a verifiable quantum secret sharing (VQSS) protocol, and a general secure multiparty quantum computation (MPQC) protocol, which can tolerate any ⌊ n−1 2 ⌋ cheaters among n players. Previous protocols for these tasks tolerated ⌊ n−1 n−1 4 ⌋ and ⌊ 6 ⌋ cheaters, respectively. The threshold we achieve is tight — even in the classical case, “fair ” multiparty computation is not possible if any set of n/2 players can cheat. Our protocols rely on approximate quantum errorcorrecting codes, which can tolerate a larger fraction of errors than traditional, exact codes. We introduce new families of authentication schemes and approximate codes tailored to the needs of our protocols, as well as new state purification techniques along the lines of those used in fault-tolerant quantum circuits.
We consider the task of sharing a secret quantum state in a quantum network in a verifiable way. We propose a protocol that achieves this task, while reducing the number of required qubits, as compared to the existing protocols. To achieve this, we combine classical encryption of the quantum secret with an existing verifiable quantum secret sharing scheme based on Calderbank-Shor-Steane quantum error correcting codes. In this way we obtain a verifiable hybrid secret sharing scheme for sharing qubits, which combines the benefits of quantum and classical schemes. Our scheme does not reveal any information to any group of less than half of the nnn nodes participating in the protocol. Moreover, for sharing a one-qubit state each node needs a quantum memory to store nnn single-qubit shares, and requires a workspace of at most 3n3n3n qubits in total to verify the quantum secret. Importantly, in our scheme an individual share is encoded in a single qubit, as opposed to previous schemes requiring $\mathrm{\ensuremath{\Omega}}(logn)$ qubits per share. Furthermore, we define a ramp verifiable hybrid scheme. We give explicit examples of various verifiable hybrid schemes based on existing quantum error correcting codes.
Zero-level CCZCCZCCZ Distillation
Magic state distillation is a key component of fault-tolerant quantum computation, as it enables the implementation of non-Clifford gates such as the TTT gate and the CCZCCZCCZ gate via gate teleportation. However, conventional distillation protocols require a large number of logical qubits and introduce substantial spatial and temporal overhead, posing a significant bottleneck for scalable fault-tolerant quantum computation. In this work, we propose a zero-level distillation protocol that efficiently generates a high-fidelity logical CCZCCZCCZ magic state using only physical qubits on a two-dimensional square lattice with nearest-neighbor interactions. Our method leverages the transversal T/T†T/T^\daggerT/T† operation of the [[8,3,2]][[ 8,3,2 ]][[8,3,2]] code to fault-tolerantly encode the state CCZ‾∣+++⟩\overline{CCZ}|+++\rangleCCZ∣+++⟩, which is subsequently teleported to three surface-code logical qubits via lattice surgery. To enable teleportation between codes with different distances, we introduce adaptively initialized teleportation (AIT), a tailored initialization procedure for the surface code. Numerical simulations demonstrate that the logical error rate scales as pL≃300×p2p_L \simeq 300 \times p^2pL​≃300×p2 with respect to the physical error rate ppp. For example, the proposed method improves the logical error rate by approximately one and two orders of magnitude at p=10−3p = 10^{-3}p=10−3 and p=10−4p = 10^{-4}p=10−4, respectively, compared to conventional seven-TTT-gate approaches. The distillation circuit requires only 22 physical qubits, 3 logical qubits, and a circuit depth of 24, reducing the space-time overhead by a factor of approximately 5-10 compared to previous methods. This result highlights the practicality of CCZCCZCCZ-state distillation in early fault-tolerant quantum computation and offers a new direction toward resource-efficient physical-level magic state distillation beyond conventional TTT-state generation.
Secure multi-party quantum computation (MPQC) protocol is a versatile tool that enables error-free distributed quantum computation to a group of nnn mutually distrustful quantum nodes even when some of the quantum nodes do not follow the instructions of the protocol honestly. However, in case of the MPQC protocols built on top of the quantum error correction technique, the versatility is significantly affected by the fact that one has to choose a particular quantum error-correcting code (QECC), which immediately applies a constraint on the number of quantum nodes nnn. Therefore, in this talk, we suggest a modified MPQC protocol based on triorthogonal QECCs which applies significantly less constraint on the number of quantum nodes nnn if compared to the previously suggested MPQC protocol based on triply-even QECCs. Especially, the variety of available options in the region of a small number of quantum nodes nnn becomes important in the noisy intermediate-scale quantum (NISQ) era.