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arXiv:2203.10106

Non-local computation of quantum circuits with small light cones

Kfir Dolev, Sam Cree

Published Mar 18, 2022• 1 citation
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Abstract

The task of non-local quantum computation requires implementation of a unitary on $n$ qubits between two parties with only one round of communication, ideally with minimal pre-shared entanglement. We introduce a new protocol that makes use of the fact that port-based teleportation costs much less entanglement when done only on a small number of qubits at a time. Whereas previous protocols have entanglement cost independent of the unitary or scaling with its complexity, the cost of the new protocol scales with the non-locality of the unitary. Specifically, it takes the form $\sim n^{4V}$ with $V$ the maximum volume of a past light cone in a circuit implementing the unitary. Thus we can implement unitary circuits with $V\sim O(1)$ using polynomial entanglement, and those with $V\sim \mathrm{polylog}(n)$ using quasi-polynomial entanglement. For a general unitary circuit with $d$ layers of $k$-qubit gates $V$ is at most $k^d$, but if geometric locality is imposed it is at most polynomial in $d$. We give an explicit class of unitaries for which our protocol's entanglement cost scales better than any known protocol. We also show that several extensions can be made without significantly affecting the entanglement cost - arbitrary local pre- and post-processing; global Clifford pre- and post-processing; and the addition of a polynomial number of auxiliary systems.

DOI
10.48550/arxiv.2203.10106

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The task of non-local quantum computation requires implementation of a unitary on nnn qubits between two parties with only one round of communication, ideally with minimal pre-shared entanglement. We introduce a new protocol that makes use of the fact that port-based teleportation costs much less entanglement when done only on a small number of qubits at a time. Whereas previous protocols have entanglement cost independent of the unitary or scaling with its complexity, the cost of the new protocol scales with the non-locality of the unitary. Specifically, it takes the form ∼n4V\sim n^{4V}∼n4V with VVV the maximum volume of a past light cone in a circuit implementing the unitary. Thus we can implement unitary circuits with V∼O(1)V\sim O(1)V∼O(1) using polynomial entanglement, and those with V∼polylog(n)V\sim \mathrm{polylog}(n)V∼polylog(n) using quasi-polynomial entanglement. For a general unitary circuit with ddd layers of kkk-qubit gates VVV is at most kdk^dkd, but if geometric locality is imposed it is at most polynomial in ddd. We give an explicit class of unitaries for which our protocol's entanglement cost scales better than any known protocol. We also show that several extensions can be made without significantly affecting the entanglement cost - arbitrary local pre- and post-processing; global Clifford pre- and post-processing; and the addition of a polynomial number of auxiliary systems.
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Instantaneous non-local quantum computation requires multiple parties to jointly perform a quantum operation, using pre-shared entanglement and a single round of simultaneous communication. We study this task for its close connection to position-based quantum cryptography, but it also has natural applications in the context of foundations of quantum physics and in distributed computing. The best known general construction for instantaneous non-local quantum computation requires a pre-shared state which is exponentially large in the number of qubits involved in the operation, while efficient constructions are known for very specific cases only. We partially close this gap by presenting new schemes for efficient instantaneous non-local computation of several classes of quantum circuits, using the Clifford+T gate set. Our main result is a protocol which uses entanglement exponential in the T-depth of a quantum circuit, able to perform non-local computation of quantum circuits with a (poly-)logarithmic number of layers of T gates with quasi-polynomial entanglement. Our proofs combine ideas from blind and delegated quantum computation with the garden-hose model, a combinatorial model of communication complexity which was recently introduced as a tool for studying certain schemes for quantum position verification. As an application of our results, we also present an efficient attack on a recently-proposed scheme for position verification by Chakraborty and Leverrier.
We address the question of efficient implementation of quantum protocols, with small communication and entanglement, and short depth circuit for encoding or decoding. We introduce two new methods to achieve this, the first method involving two new versions of the convex-split lemma that use much smaller amount of additional resource (in comparison to previous version) and the second method being inspired by the technique of classical correlated sampling in computer science literature. These lead to a series of new consequences, as follows. First, we consider the task of quantum decoupling, where the aim is to apply an operation on a n-qubit register so as to make it independent of an inaccessible quantum system. Many previous works achieve decoupling with the aid of a random unitary. It is known that random unitaries can be replaced by random circuits of size O(nlog n) and depth poly(log n), or unitary 2-designs based on Clifford circuits of similar size and depth. We show that given any choice of basis such as the computational basis, decoupling can be achieved by a unitary that takes basis vectors to basis vectors. Thus, the circuit acts in a `classical' manner and additionally uses O(n) catalytic qubits in maximally mixed quantum state. Our unitary performs addition and multiplication modulo a prime and hence achieves a circuit size of O(n\log n) and logarithmic depth. This shows that the circuit complexity of integer multiplication (modulo a prime) is lower bounded by the optimal circuit complexity of decoupling. Next, we construct a new one-shot entanglement-assisted protocol for quantum channel coding that achieves near-optimal communication through a given channel. Furthermore, the number of qubits of pre-shared entanglement is exponentially smaller than that used in the previous protocol that was near-optimal in communication. We also achieve similar results for the one-shot quantum state redistribution. Joint work with Rahul Jain. https://arxiv.org/abs/1809.07056
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