Recent years have witnessed a growing interest in topics at the intersection of many-body physics and complexity theory. Many-body physics aims to understand and classify emergent behavior of systems with a large number of particles, while complexity theory aims to classify computational problems based on how the time required to solve the problem scales as the problem size becomes large. In this work, we use insights from complexity theory to classify phases in interacting many-body systems. Specifically, we demonstrate a "complexity phase diagram" for the Bose-Hubbard model with long-range hopping. This shows how the complexity of simulating time evolution varies according to various parameters appearing in the problem, such as the evolution time, the particle density, and the degree of locality. We find that classification of complexity phases is closely related to upper bounds on the spread of quantum correlations, and protocols to transfer quantum information in a controlled manner. Our work motivates future studies of complexity in many-body systems and its interplay with the associated physical phenomena.

1 aMaskara, Nishad1 aDeshpande, Abhinav1 aTran, Minh, C.1 aEhrenberg, Adam1 aFefferman, Bill1 aGorshkov, Alexey, V. uhttps://arxiv.org/abs/1906.0417812409nas a2200169 45000080041000002450055000412100055000963000047001514900008001985201188600206100002312092700002012115700001912135700002312154700002512177856003712202 2018 eng d00aDynamical phase transitions in sampling complexity0 aDynamical phase transitions in sampling complexity a12 pages, 4 figures. v3: published version0 v1213 aWe make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time

We make the case for studying the complexity of approximately simulating (sampling) quantum systems for reasons beyond that of quantum computational supremacy, such as diagnosing phase transitions. We consider the sampling complexity as a function of time

A critical milestone on the path to useful quantum computers is quantum supremacy - a demonstration of a quantum computation that is prohibitively hard for classical computers. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, which we call Random Circuit Sampling (RCS). In this paper we study both the hardness and verification of RCS. While RCS was defined with experimental realization in mind, we show complexity theoretic evidence of hardness that is on par with the strongest theoretical proposals for supremacy. Specifically, we show that RCS satisfies an average-case hardness condition - computing output probabilities of typical quantum circuits is as hard as computing them in the worst-case, and therefore #P-hard. Our reduction exploits the polynomial structure in the output amplitudes of random quantum circuits, enabled by the Feynman path integral. In addition, it follows from known results that RCS satisfies an anti-concentration property, making it the first supremacy proposal with both average-case hardness and anti-concentration.

1 aBouland, Adam1 aFefferman, Bill1 aNirkhe, Chinmay1 aVazirani, Umesh uhttps://arxiv.org/abs/1803.0440205805nas a2200145 4500008004100000245004900041210004900090260001500139520537700154100002305531700002005554700002305574700002505597856003705622 2017 eng d00aComplexity of sampling as an order parameter0 aComplexity of sampling as an order parameter c2017/03/153 aWe consider the classical complexity of approximately simulating time evolution under spatially local quadratic bosonic Hamiltonians for time

We study the complexity of classically sampling from the output distribution of an Ising spin model, which can be implemented naturally in a variety of atomic, molecular, and optical systems. In particular, we construct a specific example of an Ising Hamiltonian that, after time evolution starting from a trivial initial state, produces a particular output configuration with probability very nearly proportional to the square of the permanent of a matrix with arbitrary integer entries. In a similar spirit to boson sampling, the ability to sample classically from the probability distribution induced by time evolution under this Hamiltonian would imply unlikely complexity theoretic consequences, suggesting that the dynamics of such a spin model cannot be efficiently simulated with a classical computer. Physical Ising spin systems capable of achieving problem-size instances (i.e., qubit numbers) large enough so that classical sampling of the output distribution is classically difficult in practice may be achievable in the near future. Unlike boson sampling, our current results only imply hardness of *exact* classical sampling, leaving open the important question of whether a much stronger approximate-sampling hardness result holds in this context. The latter is most likely necessary to enable a convincing experimental demonstration of quantum supremacy. As referenced in a recent paper [A. Bouland, L. Mancinska, and X. Zhang, in *Proceedings of the 31st Conference on Computational Complexity (CCC 2016)*, Leibniz International Proceedings in Informatics (Schloss Dagstuhl–Leibniz-Zentrum für Informatik, Dagstuhl, 2016)], our result completes the sampling hardness classification of two-qubit commuting Hamiltonians.

Quantum-mechanical devices have the potential to transform cryptography. Most research in this area has focused either on the information-theoretic advantages of quantum protocols or on the security of classical cryptographic schemes against quantum attacks. In this work, we initiate the study of another relevant topic: the encryption of quantum data in the computational setting. In this direction, we establish quantum versions of several fundamental classical results. First, we develop natural definitions for private-key and public-key encryption schemes for quantum data. We then define notions of semantic security and indistinguishability, and, in analogy with the classical work of Goldwasser and Micali, show that these notions are equivalent. Finally, we construct secure quantum encryption schemes from basic primitives. In particular, we show that quantum-secure one-way functions imply IND-CCA1-secure symmetric-key quantum encryption, and that quantum-secure trapdoor one-way permutations imply semantically-secure public-key quantum encryption.

1 aAlagic, Gorjan1 aBroadbent, Anne1 aFefferman, Bill1 aGagliardoni, Tommaso1 aSchaffner, Christian1 aJules, Michael, St. uhttps://link.springer.com/chapter/10.1007%2F978-3-319-49175-2_300861nas a2200121 4500008004100000245005500041210005500096260001500151520049300166100002000659700002400679856003600703 2016 eng d00aQuantum Merlin Arthur with Exponentially Small Gap0 aQuantum Merlin Arthur with Exponentially Small Gap c2016/01/083 aWe study the complexity of QMA proof systems with inverse exponentially small promise gap. We show that this class can be exactly characterized by PSPACE, the class of problems solvable with a polynomial amount of memory. As applications we show that a "precise" version of the Local Hamiltonian problem is PSPACE-complete, and give a provable setting in which the ability to prepare PEPS states is not as powerful as the ability to prepare the ground state of general Local Hamiltonians.1 aFefferman, Bill1 aLin, Cedric, Yen-Yu uhttp://arxiv.org/abs/1601.0197502209nas a2200121 4500008004100000245002700041210002400068260001500092520190500107100001902012700002002031856003602051 2016 eng d00aOn Quantum Obfuscation0 aQuantum Obfuscation c2016/02/043 aEncryption of data is fundamental to secure communication in the modern world. Beyond encryption of data lies obfuscation, i.e., encryption of functionality. It is well-known that the most powerful means of obfuscating classical programs, so-called ``black-box obfuscation',' is provably impossible [Barak et al '12]. However, several recent results have yielded candidate schemes that satisfy a definition weaker than black-box, and yet still have numerous applications. In this work, we initialize the rigorous study of obfuscating programs via quantum-mechanical means. We define notions of quantum obfuscation which encompass several natural variants. The input to the obfuscator can describe classical or quantum functionality, and the output can be a circuit description or a quantum state. The obfuscator can also satisfy one of a number of obfuscation conditions: black-box, information-theoretic black-box, indistinguishability, and best possible; the last two conditions come in three variants: perfect, statistical, and computational. We discuss many applications, including CPA-secure quantum encryption, quantum fully-homomorphic encryption, and public-key quantum money. We then prove several impossibility results, extending a number of foundational papers on classical obfuscation to the quantum setting. We prove that quantum black-box obfuscation is impossible in a setting where adversaries can possess more than one output of the obfuscator. In particular, generic transformation of quantum circuits into black-box-obfuscated quantum circuits is impossible. We also show that statistical indistinguishability obfuscation is impossible, up to an unlikely complexity-theoretic collapse. Our proofs involve a new tool: chosen-ciphertext-secure encryption of quantum data, which was recently shown to be possible assuming quantum-secure one-way functions exist [Alagic et al '16].1 aAlagic, Gorjan1 aFefferman, Bill uhttp://arxiv.org/abs/1602.0177121161nas a2200205 45000080041000000200022000410220014000632450069000772100068001462600015002143000016002294900007002455202053400252100002020786700002420806700002420830700002220854700002520876856005420901 2016 eng d a978-3-95977-013-2 a1868-896900aSpace-Efficient Error Reduction for Unitary Quantum Computations0 aSpaceEfficient Error Reduction for Unitary Quantum Computations c2016/04/27 a14:1--14:140 v553 aThis paper develops general space-efficient methods for error reduction for unitary quantum computation. Consider a polynomial-time quantum computation with completeness