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Quantum Verification of NP Problems with Single Photons and Linear Optics Zhang et al. Light: Science & Applications (2021) 10:169 Official journal of the CIOMP 2047-7538 https://doi.org/10.1038/s41377-021-00608-4 www.nature.com/lsa ARTICLE Open Access Quantum verification of NP problems with single photons and linear optics ✉ Aonan Zhang 1,2, Hao Zhan1,2, Junjie Liao1,2, Kaimin Zheng1,2,TaoJiang1,2,MinghaoMi1,2, Penghui Yao3 and ✉ Lijian Zhang 1,2 Abstract 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 pffiffiffi verification task with quadratically reduced information about the solution in O~ð nÞ 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. 1234567890():,; 1234567890():,; 1234567890():,; 1234567890():,; Introduction freedom of photons19,20 and well-developed linear – Quantum computing has been found to unprecedent- optics21 24, information can be encoded and processed in edly speed-up classically intractable computational a high-dimensional Hilbert space. These features make – tasks1 7. As building universal, error-corrected quantum photonics a suitable platform to realize quantum algo- computers is still challenging, the community now seeks rithms involving high-dimensional encoding, low degree practical uses of noisy intermediate-scale quantum of entanglement, and linear operations. Here we exploit (NISQ) technologies in computational problems of the advantages of photonics to realize a new regime of interest and importance5. Photonics has been a versatile quantum algorithm—the quantum verification machine – tool in quantum information tasks8 10 such as boson (QVM) of nondeterministic polynomial-time (NP) – sampling7,11 14, quantum walk9,15,16, and variational problems. quantum simulation17,18. By utilizing multi-degrees of The complexity class NP, which is the set of decision problems verifiable in polynomial time by a deterministic Turing machine, encompasses many natural decision and Correspondence: Penghui Yao ([email protected])or optimization problems. By definition, NP can be Lijian Zhang ([email protected]) abstracted as a proof system which models computation 1National Laboratory of Solid State Microstructures, Key Laboratory of Intelligent Optical Sensing and Manipulation (Ministry of Education) and as exchange of messages between the prover and the College of Engineering and Applied Sciences, Nanjing University, 210093 verifier. Verifying the correctness of a proof is a founda- Nanjing, China tional computational model underpinning both the 2Collaborative Innovation Center of Advanced Microstructures, Nanjing University, 210093 Nanjing, China complexity theory and applications such as delegated Full list of author information is available at the end of the article © The Author(s) 2021 Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a linktotheCreativeCommons license, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this license, visit http://creativecommons.org/licenses/by/4.0/. Zhang et al. Light: Science & Applications (2021) 10:169 Page 2 of 11 computation. Specifically, we focus on the verification of computationally bounded verifier Arthur to convince the first discovered and most extensively studied NP- Arthur the instance is satisfiable (see Fig. 1a). Arthur complete problem—the Boolean satisfiability problem checks the proof in his computing machines and decide (SAT)25, that is, the problem of asking whether a given whether to accept or reject the proof. Two properties are Boolean formula with n variables has a satisfying assign- required in a QMA protocol: (i) Completeness: if the ment. The NP-completeness signifies that any NP pro- instance is satisfiable, there exist a proof such that Arthur blem can be efficiently reduced to this problem. accepts with at least some high probability c; (ii) Sound- Corresponding to the problem of satisfying potentially ness: if the instance is not satisfiable, for any proof Arthur conflict constraints, SAT has found numerous applica- accepts with at most some probability s. tions in circuit design, mode checking, automated proving The protocol firstly reduces the 3-SAT instance to a 2- and artificial intelligence26. Under the widely believed out-of-4 SAT instance where each clause contains four 27 exponential time hypothesis (ETH) , which asserts that variables xi,xj,xk,xl and is satisfied if two of them are true, the best algorithm for solving 3-SAT (a representative i.e., xi + xj + xk_xl = 2. In the verification, Merlins are γ pffiffiffi form of SAT) runs in time 2 n for some constant γ >0, supposed to send Arthur K ¼ Oð nÞ identical, unen- verifying 3-SAT requires at least O(n) bits. Otherwise the tangled quantum states28, each of the form verifier can simply enumerate overall possible proofs, X which yield a sub-exponential algorithm for solving 1 n x jψi¼pffiffiffi ðÀ1Þ i jii ð1Þ 3-SAT. Surprisingly, this bound on proof length no longer n i¼1 applies if quantum bits are used in proofs and verified by ^y ^y quantum computers. This perception rapidly aroused where jii¼ai j0i and ai is the creation operator on mode i. n substantial efforts on quantum verification of NP(-com- Here x1,x2,...,xn ∈ {0,1} is an assignment of the n variables. – plete) problems28 35. In this line, Aaronson et al. pro- A state of such form is called a proper state. The n- pffiffiffi posed a protocol of proving 3-SAT with Oð nÞ dimensional quantum state can be equivalently described by unentangled quantum states each of O(logn) qubits28 and logn qubits revealing at most logn bits information by variants of the protocol have also been developed30,32. measurements on the state. To check whether the assign- However, to date a complete demonstration of quantum ment x satisfies the clauses, Arthur can choose some clauses verification algorithm is still missing. (i,j,k,l) at random and measure the K copies of |ψ〉 in a basis In this work, we report the first experimental quantum withaprojectionon|c〉 = (|i〉 + |j〉 + |k〉 + |l〉)/2 for each verification of SAT with single photons and linear optics, clause. For each copy Arthur will get a probability of by implementing a modified version of recent proposals34. observing the outcome |c〉 We present a scalable design of reconfigurable optical 2 xi xj xk xl 2 circuits in which quantum proofs are mapped to single pc ¼jhcjψij ¼ ½ðÀ1Þ þðÀ1Þ þðÀ1Þ þðÀ1Þ =4n photons distributed in optical modes. The experiment demonstrates faithful verification of NP problems in Then Arthur rejects the proof if he gets the outcome |c〉 terms of a complete analysis on the satisfiable instance, for at least one copy and accepts it otherwise. With this unsatisfiable instance and cheating prover cases. Our Satisfiability Test, Arthur will have pc = 0ifxi + xj + xk + work links the remarkable proof systems in computer xl = 2, and some constant nonzero probability otherwise. science to the manipulation and detection of photons, An issue is that Merlins may cheat Arthur by sending him which foreshadows further investigations of a variety of improper state, for example concentrating the amplitude computational models in the photonic regime. in a subset of the basis {|i〉} such that the Satisfiability Test passes even the instance is not satisfiable. To tackle Results this problem Arthur can perform Uniformity Test:he Quantum verification algorithm of the satisfiability randomly chooses a matching M on the set {1,...,n} such problem that the set is partitioned into n/2 groups of the form (i,j), An instance of SAT is formalized as the conjunction of a then measures each copy of the state |ψ〉 in the basis with set of clauses ϕ = c1 ∧ c2... ∧ cj, each of which is the {|i〉 + |j〉,|i〉 − |j〉} for each (i,j) ∈ M. Only if the state is disjunction of a set of literals l1 ∨ l2... ∨ lm. A literal could proper (i.e., the amplitudes are equal), one of the two be a variable xi or a negation of a variable ¬xi. In 3-SAT outcomes will never occur. With the statistics on the instances, each clause has exactly three literals. The outcomes, Arthur rejects the proof if two outcomes {|i〉 quantum verification of 3-SAT corresponds to the com- + |j〉,|i〉 − |j〉} both occur for a same (i,j) ∈ M. Here the K plexity class Quantum Merlin-Arthur [QMA(K)], as the copies are used to obtain sufficient statistics on the out- – quantum analogue of NP36 38.
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