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**Published in:** LIPIcs, Volume 300, 39th Computational Complexity Conference (CCC 2024)

Given a local Hamiltonian, how difficult is it to determine the entanglement structure of its ground state? We show that this problem is computationally intractable even if one is only trying to decide if the ground state is volume-law vs near area-law entangled. We prove this by constructing strong forms of pseudoentanglement in a public-key setting, where the circuits used to prepare the states are public knowledge. In particular, we construct two families of quantum circuits which produce volume-law vs near area-law entangled states, but nonetheless the classical descriptions of the circuits are indistinguishable under the Learning with Errors (LWE) assumption. Indistinguishability of the circuits then allows us to translate our construction to Hamiltonians. Our work opens new directions in Hamiltonian complexity, for example whether it is difficult to learn certain phases of matter.

Adam Bouland, Bill Fefferman, Soumik Ghosh, Tony Metger, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure. In 39th Computational Complexity Conference (CCC 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 300, pp. 21:1-21:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2024.21, author = {Bouland, Adam and Fefferman, Bill and Ghosh, Soumik and Metger, Tony and Vazirani, Umesh and Zhang, Chenyi and Zhou, Zixin}, title = {{Public-Key Pseudoentanglement and the Hardness of Learning Ground State Entanglement Structure}}, booktitle = {39th Computational Complexity Conference (CCC 2024)}, pages = {21:1--21:23}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-331-7}, ISSN = {1868-8969}, year = {2024}, volume = {300}, editor = {Santhanam, Rahul}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2024.21}, URN = {urn:nbn:de:0030-drops-204175}, doi = {10.4230/LIPIcs.CCC.2024.21}, annote = {Keywords: Quantum computing, Quantum complexity theory, entanglement} }

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**Published in:** LIPIcs, Volume 287, 15th Innovations in Theoretical Computer Science Conference (ITCS 2024)

Entanglement is a quantum resource, in some ways analogous to randomness in classical computation. Inspired by recent work of Gheorghiu and Hoban, we define the notion of "pseudoentanglement", a property exhibited by ensembles of efficiently constructible quantum states which are indistinguishable from quantum states with maximal entanglement. Our construction relies on the notion of quantum pseudorandom states - first defined by Ji, Liu and Song - which are efficiently constructible states indistinguishable from (maximally entangled) Haar-random states. Specifically, we give a construction of pseudoentangled states with entanglement entropy arbitrarily close to log n across every cut, a tight bound providing an exponential separation between computational vs information theoretic quantum pseudorandomness. We discuss applications of this result to Matrix Product State testing, entanglement distillation, and the complexity of the AdS/CFT correspondence. As compared with a previous version of this manuscript (arXiv:2211.00747v1) this version introduces a new pseudorandom state construction, has a simpler proof of correctness, and achieves a technically stronger result of low entanglement across all cuts simultaneously.

Scott Aaronson, Adam Bouland, Bill Fefferman, Soumik Ghosh, Umesh Vazirani, Chenyi Zhang, and Zixin Zhou. Quantum Pseudoentanglement. In 15th Innovations in Theoretical Computer Science Conference (ITCS 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 287, pp. 2:1-2:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)

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@InProceedings{aaronson_et_al:LIPIcs.ITCS.2024.2, author = {Aaronson, Scott and Bouland, Adam and Fefferman, Bill and Ghosh, Soumik and Vazirani, Umesh and Zhang, Chenyi and Zhou, Zixin}, title = {{Quantum Pseudoentanglement}}, booktitle = {15th Innovations in Theoretical Computer Science Conference (ITCS 2024)}, pages = {2:1--2:21}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-309-6}, ISSN = {1868-8969}, year = {2024}, volume = {287}, editor = {Guruswami, Venkatesan}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2024.2}, URN = {urn:nbn:de:0030-drops-195300}, doi = {10.4230/LIPIcs.ITCS.2024.2}, annote = {Keywords: Quantum computing, Quantum complexity theory, entanglement} }

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Abstract

**Published in:** LIPIcs, Volume 151, 11th Innovations in Theoretical Computer Science Conference (ITCS 2020)

The AdS/CFT correspondence is central to efforts to reconcile gravity and quantum mechanics, a fundamental goal of physics. It posits a duality between a gravitational theory in Anti de Sitter (AdS) space and a quantum mechanical conformal field theory (CFT), embodied in a map known as the AdS/CFT dictionary mapping states to states and operators to operators. This dictionary map is not well understood and has only been computed on special, structured instances. In this work we introduce cryptographic ideas to the study of AdS/CFT, and provide evidence that either the dictionary must be exponentially hard to compute, or else the quantum Extended Church-Turing thesis must be false in quantum gravity.
Our argument has its origins in a fundamental paradox in the AdS/CFT correspondence known as the wormhole growth paradox. The paradox is that the CFT is believed to be "scrambling" - i.e. the expectation value of local operators equilibrates in polynomial time - whereas the gravity theory is not, because the interiors of certain black holes known as "wormholes" do not equilibrate and instead their volume grows at a linear rate for at least an exponential amount of time. So what could be the CFT dual to wormhole volume? Susskind’s proposed resolution was to equate the wormhole volume with the quantum circuit complexity of the CFT state. From a computer science perspective, circuit complexity seems like an unusual choice because it should be difficult to compute, in contrast to physical quantities such as wormhole volume.
We show how to create pseudorandom quantum states in the CFT, thereby arguing that their quantum circuit complexity is not "feelable", in the sense that it cannot be approximated by any efficient experiment. This requires a specialized construction inspired by symmetric block ciphers such as DES and AES, since unfortunately existing constructions based on quantum-resistant one way functions cannot be used in the context of the wormhole growth paradox as only very restricted operations are allowed in the CFT. By contrast we argue that the wormhole volume is "feelable" in some general but non-physical sense. The duality between a "feelable" quantity and an "unfeelable" quantity implies that some aspect of this duality must have exponential complexity. More precisely, it implies that either the dictionary is exponentially complex, or else the quantum gravity theory is exponentially difficult to simulate on a quantum computer.
While at first sight this might seem to justify the discomfort of complexity theorists with equating computational complexity with a physical quantity, a further examination of our arguments shows that any resolution of the wormhole growth paradox must equate wormhole volume to an "unfeelable" quantity, leading to the same conclusions. In other words this discomfort is an inevitable consequence of the paradox.

Adam Bouland, Bill Fefferman, and Umesh Vazirani. Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality (Abstract). In 11th Innovations in Theoretical Computer Science Conference (ITCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 151, pp. 63:1-63:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020)

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@InProceedings{bouland_et_al:LIPIcs.ITCS.2020.63, author = {Bouland, Adam and Fefferman, Bill and Vazirani, Umesh}, title = {{Computational Pseudorandomness, the Wormhole Growth Paradox, and Constraints on the AdS/CFT Duality}}, booktitle = {11th Innovations in Theoretical Computer Science Conference (ITCS 2020)}, pages = {63:1--63:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-134-4}, ISSN = {1868-8969}, year = {2020}, volume = {151}, editor = {Vidick, Thomas}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2020.63}, URN = {urn:nbn:de:0030-drops-117486}, doi = {10.4230/LIPIcs.ITCS.2020.63}, annote = {Keywords: Quantum complexity theory, pseudorandomness, AdS/CFT correspondence} }

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**Published in:** LIPIcs, Volume 124, 10th Innovations in Theoretical Computer Science Conference (ITCS 2019)

A critical goal for the field of quantum computation is quantum supremacy - a demonstration of any quantum computation that is prohibitively hard for classical computers. It is both a necessary milestone on the path to useful quantum computers as well as a test of quantum theory in the realm of high complexity. A leading near-term candidate, put forth by the Google/UCSB team, is sampling from the probability distributions of randomly chosen quantum circuits, called Random Circuit Sampling (RCS).
While RCS was defined with experimental realization in mind, we give strong complexity-theoretic evidence for the classical hardness of RCS, placing it on par with the best 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 also satisfies an anti-concentration property, namely that errors in estimating output probabilities are small with respect to the probabilities themselves. This makes RCS the first proposal for quantum supremacy with both of these properties. We also give a natural condition under which an existing statistical measure, cross-entropy, verifies RCS, as well as describe a new verification measure which in some formal sense maximizes the information gained from experimental samples.

Adam Bouland, Bill Fefferman, Chinmay Nirkhe, and Umesh Vazirani. "Quantum Supremacy" and the Complexity of Random Circuit Sampling. In 10th Innovations in Theoretical Computer Science Conference (ITCS 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 124, pp. 15:1-15:2, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

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@InProceedings{bouland_et_al:LIPIcs.ITCS.2019.15, author = {Bouland, Adam and Fefferman, Bill and Nirkhe, Chinmay and Vazirani, Umesh}, title = {{"Quantum Supremacy" and the Complexity of Random Circuit Sampling}}, booktitle = {10th Innovations in Theoretical Computer Science Conference (ITCS 2019)}, pages = {15:1--15:2}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-095-8}, ISSN = {1868-8969}, year = {2019}, volume = {124}, editor = {Blum, Avrim}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2019.15}, URN = {urn:nbn:de:0030-drops-101084}, doi = {10.4230/LIPIcs.ITCS.2019.15}, annote = {Keywords: quantum supremacy, average-case hardness, verification} }

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**Published in:** LIPIcs, Volume 111, 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)

The Solovay-Kitaev theorem states that universal quantum gate sets can be exchanged with low overhead. More specifically, any gate on a fixed number of qudits can be simulated with error epsilon using merely polylog(1/epsilon) gates from any finite universal quantum gate set G. One drawback to the theorem is that it requires the gate set G to be closed under inversion. Here we show that this restriction can be traded for the assumption that G contains an irreducible representation of any finite group G. This extends recent work of Sardharwalla et al. [Sardharwalla et al., 2016], and applies also to gates from the special linear group. Our work can be seen as partial progress towards the long-standing open problem of proving an inverse-free Solovay-Kitaev theorem [Dawson and Nielsen, 2006; Kuperberg, 2015].

Adam Bouland and Maris Ozols. Trading Inverses for an Irrep in the Solovay-Kitaev Theorem. In 13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 111, pp. 6:1-6:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bouland_et_al:LIPIcs.TQC.2018.6, author = {Bouland, Adam and Ozols, Maris}, title = {{Trading Inverses for an Irrep in the Solovay-Kitaev Theorem}}, booktitle = {13th Conference on the Theory of Quantum Computation, Communication and Cryptography (TQC 2018)}, pages = {6:1--6:15}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-080-4}, ISSN = {1868-8969}, year = {2018}, volume = {111}, editor = {Jeffery, Stacey}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.TQC.2018.6}, URN = {urn:nbn:de:0030-drops-92539}, doi = {10.4230/LIPIcs.TQC.2018.6}, annote = {Keywords: Solovay-Kitaev theorem, quantum gate sets, gate set compilation} }

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**Published in:** LIPIcs, Volume 102, 33rd Computational Complexity Conference (CCC 2018)

Clifford circuits - i.e. circuits composed of only CNOT, Hadamard, and pi/4 phase gates - play a central role in the study of quantum computation. However, their computational power is limited: a well-known result of Gottesman and Knill states that Clifford circuits are efficiently classically simulable. We show that in contrast, "conjugated Clifford circuits" (CCCs) - where one additionally conjugates every qubit by the same one-qubit gate U - can perform hard sampling tasks. In particular, we fully classify the computational power of CCCs by showing that essentially any non-Clifford conjugating unitary U can give rise to sampling tasks which cannot be efficiently classically simulated to constant multiplicative error, unless the polynomial hierarchy collapses. Furthermore, by standard techniques, this hardness result can be extended to allow for the more realistic model of constant additive error, under a plausible complexity-theoretic conjecture. This work can be seen as progress towards classifying the computational power of all restricted quantum gate sets.

Adam Bouland, Joseph F. Fitzsimons, and Dax Enshan Koh. Complexity Classification of Conjugated Clifford Circuits. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 21:1-21:25, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2018.21, author = {Bouland, Adam and Fitzsimons, Joseph F. and Koh, Dax Enshan}, title = {{Complexity Classification of Conjugated Clifford Circuits}}, booktitle = {33rd Computational Complexity Conference (CCC 2018)}, pages = {21:1--21:25}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-069-9}, ISSN = {1868-8969}, year = {2018}, volume = {102}, editor = {Servedio, Rocco A.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2018.21}, URN = {urn:nbn:de:0030-drops-88677}, doi = {10.4230/LIPIcs.CCC.2018.21}, annote = {Keywords: gate set classification, quantum advantage, sampling problems, polynomial hierarchy} }

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**Published in:** LIPIcs, Volume 58, 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)

What is the minimum amount of information and time needed to solve 2SAT? When the instance is known, it can be solved in polynomial time, but is this also possible without knowing the instance? Bei, Chen and Zhang (STOC'13) considered a model where the input is accessed by proposing possible assignments to a special oracle. This oracle, on encountering some constraint unsatisfied by the proposal, returns only the constraint index. It turns out that, in this model, even 1SAT cannot be solved in polynomial time unless P=NP. Hence, we consider a model in which the input is accessed by proposing probability distributions over assignments to the variables. The oracle then returns the index of the constraint that is most likely to be violated by this distribution. We show that the information obtained this way is sufficient to solve 1SAT in polynomial time, even when the clauses can be repeated. For 2SAT, as long as there are no repeated clauses, in polynomial time we can even learn an equivalent formula for the hidden instance and hence also solve it. Furthermore, we extend these results to the quantum regime. We show that in this setting 1QSAT can be solved in polynomial time up to constant precision, and 2QSAT can be learnt in polynomial time up to inverse polynomial precision.

Itai Arad, Adam Bouland, Daniel Grier, Miklos Santha, Aarthi Sundaram, and Shengyu Zhang. On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems. In 41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 58, pp. 12:1-12:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{arad_et_al:LIPIcs.MFCS.2016.12, author = {Arad, Itai and Bouland, Adam and Grier, Daniel and Santha, Miklos and Sundaram, Aarthi and Zhang, Shengyu}, title = {{On the Complexity of Probabilistic Trials for Hidden Satisfiability Problems}}, booktitle = {41st International Symposium on Mathematical Foundations of Computer Science (MFCS 2016)}, pages = {12:1--12:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-016-3}, ISSN = {1868-8969}, year = {2016}, volume = {58}, editor = {Faliszewski, Piotr and Muscholl, Anca and Niedermeier, Rolf}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.MFCS.2016.12}, URN = {urn:nbn:de:0030-drops-64284}, doi = {10.4230/LIPIcs.MFCS.2016.12}, annote = {Keywords: computational complexity, satisfiability problems, trial and error, quantum computing, learning theory} }

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**Published in:** LIPIcs, Volume 50, 31st Conference on Computational Complexity (CCC 2016)

We classify two-qubit commuting Hamiltonians in terms of their computational complexity. Suppose one has a two-qubit commuting Hamiltonian H which one can apply to any pair of qubits, starting in a computational basis state. We prove a dichotomy theorem: either this model is efficiently classically simulable or it allows one to sample from probability distributions which cannot be sampled from classically unless the polynomial hierarchy collapses. Furthermore, the only simulable Hamiltonians are those which fail to generate entanglement. This shows that generic two-qubit commuting Hamiltonians can be used to perform computational tasks which are intractable for classical computers under plausible assumptions. Our proof makes use of new postselection gadgets and Lie theory.

Adam Bouland, Laura Mancinska, and Xue Zhang. Complexity Classification of Two-Qubit Commuting Hamiltonians. In 31st Conference on Computational Complexity (CCC 2016). Leibniz International Proceedings in Informatics (LIPIcs), Volume 50, pp. 28:1-28:33, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2016)

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@InProceedings{bouland_et_al:LIPIcs.CCC.2016.28, author = {Bouland, Adam and Mancinska, Laura and Zhang, Xue}, title = {{Complexity Classification of Two-Qubit Commuting Hamiltonians}}, booktitle = {31st Conference on Computational Complexity (CCC 2016)}, pages = {28:1--28:33}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-008-8}, ISSN = {1868-8969}, year = {2016}, volume = {50}, editor = {Raz, Ran}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CCC.2016.28}, URN = {urn:nbn:de:0030-drops-58469}, doi = {10.4230/LIPIcs.CCC.2016.28}, annote = {Keywords: Quantum Computing, Sampling Problems, Commuting Hamiltonians, IQP, Gate Classification Theorems} }