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International Conference on Wireless Communications and Signal Processing (WCSP)

Experimental verification of NP-complete problems via linear optics

Jian Li, Tongjun Liu, Tianlei Hou, Xiaorun Wang, Chenxi Liu, Pan Dai, Qin Wang

Published Oct 20, 2021• 1 citation
Publisher

Abstract

Non-deterministic polynomial (NP)-complete problems are collections of intractable problems for classical computers, but they may be easier to solve for quantum computers. Considering that practical quantum computers and networks are still under development, it is necessary to find alternatives to solve these problems. Simulations with linear optics offer a good choice since much of the research on quantum information is executed with this approach. In this work, based on Aaronson's protocol (Theory Comput, 5, 1, 2009) ad Arrazola et. al.'s linear optics scheme (npj Quantum Inf, 78, 022310, 2018), we present a demonstration of the NP-complete 2-out-of-4 SAT problem for 4 variables. The experimental results show that NP-complete problems can be well simulated through linear optical systems.

DOI
10.1109/wcsp52459.2021.9613248

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Non-deterministic polynomial (NP)-complete problems are collections of intractable problems for classical computers, but they may be easier to solve for quantum computers. Considering that practical quantum computers and networks are still under development, it is necessary to find alternatives to solve these problems. Simulations with linear optics offer a good choice since much of the research on quantum information is executed with this approach. In this work, based on Aaronson's protocol (Theory Comput, 5, 1, 2009) ad Arrazola et. al.'s linear optics scheme (npj Quantum Inf, 78, 022310, 2018), we present a demonstration of the NP-complete 2-out-of-4 SAT problem for 4 variables. The experimental results show that NP-complete problems can be well simulated through linear optical systems.
Quantum computing is seeking to realize hardware-optimized algorithms for application-related computational tasks. NP (nondeterministic-polynomial-time) is a complexity class containing many important but intractable problems like the satisfiability of potentially conflict constraints (SAT). According to the well-founded exponential time hypothesis, verifying an SAT instance of size n requires generally the complete solution in an O(n)-bit proof. In contrast, quantum verification algorithms, which encode the solution into quantum bits rather than classical bit strings, can perform the verification task with quadratically reduced information about the solution in [Formula: see text] qubits. Here we realize the quantum verification machine of SAT with single photons and linear optics. By using tunable optical setups, we efficiently verify satisfiable and unsatisfiable SAT instances and achieve a clear completeness-soundness gap even in the presence of experimental imperfections. The protocol requires only unentangled photons, linear operations on multiple modes and at most two-photon joint measurements. These features make the protocol suitable for photonic realization and scalable to large problem sizes with the advances in high-dimensional quantum information manipulation and large scale linear-optical systems. Our results open an essentially new route toward quantum advantages and extend the computational capability of optical quantum computing.
Abstract Demonstrating quantum superiority for some computational task will be a milestone for quantum technologies and would show that computational advantages are possible not only with a universal quantum computer but with simpler physical devices. Linear optics is such a simpler but powerful platform where classically-hard information processing tasks, such as Boson Sampling, can be in principle implemented. In this work, we study a fundamentally different type of computational task to achieve quantum superiority using linear optics, namely the task of verifying NP-complete problems. We focus on a protocol by Aaronson et al. (2008) that uses quantum proofs for verification. We show that the proof states can be implemented in terms of a single photon in an equal superposition over many optical modes. Similarly, the tests can be performed using linear-optical transformations consisting of a few operations: a global permutation of all modes, simple interferometers acting on at most four modes, and measurement using single-photon detectors. We also show that the protocol can tolerate experimental imperfections.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P^#P=BPP^NP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation. Our main result suggests that even an approximate or noisy classical simulation would already imply a collapse of the polynomial hierarchy. For this, we need two unproven conjectures: the "Permanent-of-Gaussians Conjecture", which says that it is #P-hard to approximate the permanent of a matrix A of independent N(0,1) Gaussian entries, with high probability over A; and the "Permanent Anti-Concentration Conjecture", which says that |Per(A)|>=sqrt(n!)/poly(n) with high probability over A. We present evidence for these conjectures, both of which seem interesting even apart from our application. This paper does not assume knowledge of quantum optics. Indeed, part of its goal is to develop the beautiful theory of noninteracting bosons underlying our model, and its connection to the permanent function, in a self-contained way accessible to theoretical computer scientists.
In recent years, many computational tasks have been proposed as candidates for showing a quantum computational advantage, that is an advantage in the time needed to perform the task using a quantum instead of a classical machine. Nevertheless, practical demonstrations of such an advantage remain particularly challenging because of the difficulty in bringing together all necessary theoretical and experimental ingredients. Here, we show an experimental demonstration of a quantum computational advantage in a prover-verifier interactive setting, where the computational task consists in the verification of an NP-complete problem by a verifier who only gets limited information about the proof sent by an untrusted prover in the form of a series of unentangled quantum states. We provide a simple linear optical implementation that can perform this verification task efficiently (within a few seconds), while we also provide strong evidence that, fixing the size of the proof, a classical computer would take much longer time (assuming only that it takes exponential time to solve an NP-complete problem). While our computational advantage concerns a specific task in a scenario of mostly theoretical interest, it brings us a step closer to potential useful applications, such as server-client quantum computing.
We give new evidence that quantum computers -- moreover, rudimentary quantum computers built entirely out of linear-optical elements -- cannot be efficiently simulated by classical computers. In particular, we define a model of computation in which identical photons are generated, sent through a linear-optical network, then nonadaptively measured to count the number of photons in each mode. This model is not known or believed to be universal for quantum computation, and indeed, we discuss the prospects for realizing the model using current technology. On the other hand, we prove that the model is able to solve sampling problems and search problems that are classically intractable under plausible assumptions. Our first result says that, if there exists a polynomial-time classical algorithm that samples from the same probability distribution as a linear-optical network, then P#P=BPPNP, and hence the polynomial hierarchy collapses to the third level. Unfortunately, this result assumes an extremely accurate simulation.