Document

**Published in:** LIPIcs, Volume 215, 13th Innovations in Theoretical Computer Science Conference (ITCS 2022)

Whereas quantum complexity theory has traditionally been concerned with problems arising from classical complexity theory (such as computing boolean functions), it also makes sense to study the complexity of inherently quantum operations such as constructing quantum states or performing unitary transformations. With this motivation, we define models of interactive proofs for synthesizing quantum states and unitaries, where a polynomial-time quantum verifier interacts with an untrusted quantum prover, and a verifier who accepts also outputs an approximation of the target state (for the state synthesis problem) or the result of the target unitary applied to the input state (for the unitary synthesis problem); furthermore there should exist an "honest" prover which the verifier accepts with probability 1.
Our main result is a "state synthesis" analogue of the inclusion PSPACE ⊆ IP: any sequence of states computable by a polynomial-space quantum algorithm (which may run for exponential time) admits an interactive protocol of the form described above. Leveraging this state synthesis protocol, we also give a unitary synthesis protocol for polynomial space-computable unitaries that act nontrivially on only a polynomial-dimensional subspace. We obtain analogous results in the setting with multiple entangled provers as well.

Gregory Rosenthal and Henry Yuen. Interactive Proofs for Synthesizing Quantum States and Unitaries. In 13th Innovations in Theoretical Computer Science Conference (ITCS 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 215, pp. 112:1-112:4, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

Copy BibTex To Clipboard

@InProceedings{rosenthal_et_al:LIPIcs.ITCS.2022.112, author = {Rosenthal, Gregory and Yuen, Henry}, title = {{Interactive Proofs for Synthesizing Quantum States and Unitaries}}, booktitle = {13th Innovations in Theoretical Computer Science Conference (ITCS 2022)}, pages = {112:1--112:4}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-217-4}, ISSN = {1868-8969}, year = {2022}, volume = {215}, editor = {Braverman, Mark}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2022.112}, URN = {urn:nbn:de:0030-drops-157086}, doi = {10.4230/LIPIcs.ITCS.2022.112}, annote = {Keywords: interactive proofs, quantum state complexity, quantum unitary complexity} }

Document

**Published in:** LIPIcs, Volume 185, 12th Innovations in Theoretical Computer Science Conference (ITCS 2021)

QAC circuits are quantum circuits with one-qubit gates and Toffoli gates of arbitrary arity. QAC^0 circuits are QAC circuits of constant depth, and are quantum analogues of AC^0 circuits. We prove the following:
- For all d ≥ 7 and ε > 0 there is a depth-d QAC circuit of size exp(poly(n^{1/d}) log(n/ε)) that approximates the n-qubit parity function to within error ε on worst-case quantum inputs. Previously it was unknown whether QAC circuits of sublogarithmic depth could approximate parity regardless of size.
- We introduce a class of "mostly classical" QAC circuits, including a major component of our circuit from the above upper bound, and prove a tight lower bound on the size of low-depth, mostly classical QAC circuits that approximate this component.
- Arbitrary depth-d QAC circuits require at least Ω(n/d) multi-qubit gates to achieve a 1/2 + exp(-o(n/d)) approximation of parity. When d = Θ(log n) this nearly matches an easy O(n) size upper bound for computing parity exactly.
- QAC circuits with at most two layers of multi-qubit gates cannot achieve a 1/2 + exp(-o(n)) approximation of parity, even non-cleanly. Previously it was known only that such circuits could not cleanly compute parity exactly for sufficiently large n.
The proofs use a new normal form for quantum circuits which may be of independent interest, and are based on reductions to the problem of constructing certain generalizations of the cat state which we name "nekomata" after an analogous cat yōkai.

Gregory Rosenthal. Bounds on the QAC^0 Complexity of Approximating Parity. In 12th Innovations in Theoretical Computer Science Conference (ITCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 185, pp. 32:1-32:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)

Copy BibTex To Clipboard

@InProceedings{rosenthal:LIPIcs.ITCS.2021.32, author = {Rosenthal, Gregory}, title = {{Bounds on the QAC^0 Complexity of Approximating Parity}}, booktitle = {12th Innovations in Theoretical Computer Science Conference (ITCS 2021)}, pages = {32:1--32:20}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-177-1}, ISSN = {1868-8969}, year = {2021}, volume = {185}, editor = {Lee, James R.}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2021.32}, URN = {urn:nbn:de:0030-drops-135713}, doi = {10.4230/LIPIcs.ITCS.2021.32}, annote = {Keywords: quantum circuit complexity, QAC^0, fanout, parity, nekomata} }

Document

**Published in:** LIPIcs, Volume 148, 14th International Symposium on Parameterized and Exact Computation (IPEC 2019)

For any fixed graph G, the subgraph isomorphism problem asks whether an n-vertex input graph has a subgraph isomorphic to G. A well-known algorithm of Alon, Yuster and Zwick (1995) efficiently reduces this to the "colored" version of the problem, denoted G-SUB, and then solves G-SUB in time O(n^{tw(G)+1}) where tw(G) is the treewidth of G. Marx (2010) conjectured that G-SUB requires time Omega(n^{const * tw(G)}) and, assuming the Exponential Time Hypothesis, proved a lower bound of Omega(n^{const * emb(G)}) for a certain graph parameter emb(G) = Omega(tw(G)/log tw(G)). With respect to the size of AC^0 circuits solving G-SUB, Li, Razborov and Rossman (2017) proved an unconditional average-case lower bound of Omega(n^{kappa(G)}) for a different graph parameter kappa(G) = Omega(tw(G)/log tw(G)).
Our contributions are as follows. First, we show that emb(G) is at most O(kappa(G)) for all graphs G. Next, we show that kappa(G) can be asymptotically less than tw(G); for example, if G is a hypercube then kappa(G) is Theta(tw(G)/sqrt{log tw(G)}). Finally, we construct AC^0 circuits of size O(n^{kappa(G)+const}) that solve G-SUB in the average case, on a variety of product distributions. This improves an O(n^{2 kappa(G)+const}) upper bound of Li et al., and shows that the average-case complexity of G-SUB is n^{o(tw(G))} for certain families of graphs G such as hypercubes.

Gregory Rosenthal. Beating Treewidth for Average-Case Subgraph Isomorphism. In 14th International Symposium on Parameterized and Exact Computation (IPEC 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 148, pp. 24:1-24:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019)

Copy BibTex To Clipboard

@InProceedings{rosenthal:LIPIcs.IPEC.2019.24, author = {Rosenthal, Gregory}, title = {{Beating Treewidth for Average-Case Subgraph Isomorphism}}, booktitle = {14th International Symposium on Parameterized and Exact Computation (IPEC 2019)}, pages = {24:1--24:14}, series = {Leibniz International Proceedings in Informatics (LIPIcs)}, ISBN = {978-3-95977-129-0}, ISSN = {1868-8969}, year = {2019}, volume = {148}, editor = {Jansen, Bart M. P. and Telle, Jan Arne}, publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik}, address = {Dagstuhl, Germany}, URL = {https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.IPEC.2019.24}, URN = {urn:nbn:de:0030-drops-114850}, doi = {10.4230/LIPIcs.IPEC.2019.24}, annote = {Keywords: subgraph isomorphism, average-case complexity, AC^0, circuit complexity} }

X

Feedback for Dagstuhl Publishing

Feedback submitted

Please try again later or send an E-mail