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Documents authored by Schaeffer, Luke

Document
DOI: 10.4230/LIPIcs.ITCS.2025.32
Succinct Fermion Data Structures

Authors: Joseph Carolan and Luke Schaeffer

Published in: LIPIcs, Volume 325, 16th Innovations in Theoretical Computer Science Conference (ITCS 2025)

Abstract
Simulating fermionic systems on a quantum computer requires representing fermionic states using qubits. The complexity of many simulation algorithms depends on the complexity of implementing rotations generated by fermionic creation-annihilation operators, and the space depends on the number of qubits used. While standard fermion encodings like Jordan-Wigner are space optimal for arbitrary fermionic systems, physical symmetries like particle conservation can reduce the number of physical configurations, allowing improved space complexity. Such space saving is only feasible if the gate overhead is small, suggesting a (quantum) data structures problem, wherein one would like to minimize space used to represent a fermionic state, while still enabling efficient rotations. We define a structure which naturally captures mappings from fermions to systems of qubits. We then instantiate it in two ways, giving rise to two new second-quantized fermion encodings of F fermions in M modes. An information theoretic minimum of I: = ⌈log₂ binom(M,F)⌉ qubits is required for such systems, a bound we nearly match over the entire parameter regime. 1) Our first construction uses I + o(I) qubits when F = o(M), and allows rotations generated by creation-annihilation operators in O(I) gates and O(log M log log M) depth. 2) Our second construction uses I + O(1) qubits when F = Θ(M), and allows rotations generated by creation-annihilation operators in O(I³) gates. In relation to comparable prior work, the first represents a polynomial improvement in both space and gate complexity (against Kirby et al. 2022), and the second represents an exponential improvement in gate complexity at the cost of only a constant number of additional qubits (against Harrison et al. or Shee et al. 2022), in the described parameter regimes.
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Joseph Carolan and Luke Schaeffer. Succinct Fermion Data Structures. In 16th Innovations in Theoretical Computer Science Conference (ITCS 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 325, pp. 32:1-32:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025)

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@InProceedings{carolan_et_al:LIPIcs.ITCS.2025.32,
  author =	{Carolan, Joseph and Schaeffer, Luke},
  title =	{{Succinct Fermion Data Structures}},
  booktitle =	{16th Innovations in Theoretical Computer Science Conference (ITCS 2025)},
  pages =	{32:1--32:21},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-361-4},
  ISSN =	{1868-8969},
  year =	{2025},
  volume =	{325},
  editor =	{Meka, Raghu},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2025.32},
  URN =		{urn:nbn:de:0030-drops-226600},
  doi =		{10.4230/LIPIcs.ITCS.2025.32},
  annote =	{Keywords: quantum computing, data structures, fermion encodings}
}
Document
DOI: 10.4230/LIPIcs.CSL.2022.24
Decidability for Sturmian Words

Authors: Philipp Hieronymi, Dun Ma, Reed Oei, Luke Schaeffer, Christian Schulz, and Jeffrey Shallit

Published in: LIPIcs, Volume 216, 30th EACSL Annual Conference on Computer Science Logic (CSL 2022)

Abstract
We show that the first-order theory of Sturmian words over Presburger arithmetic is decidable. Using a general adder recognizing addition in Ostrowski numeration systems by Baranwal, Schaeffer and Shallit, we prove that the first-order expansions of Presburger arithmetic by a single Sturmian word are uniformly ω-automatic, and then deduce the decidability of the theory of the class of such structures. Using an implementation of this decision algorithm called Pecan, we automatically reprove classical theorems about Sturmian words in seconds, and are able to obtain new results about antisquares and antipalindromes in characteristic Sturmian words.
Cite as

Philipp Hieronymi, Dun Ma, Reed Oei, Luke Schaeffer, Christian Schulz, and Jeffrey Shallit. Decidability for Sturmian Words. In 30th EACSL Annual Conference on Computer Science Logic (CSL 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 216, pp. 24:1-24:23, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)

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@InProceedings{hieronymi_et_al:LIPIcs.CSL.2022.24,
  author =	{Hieronymi, Philipp and Ma, Dun and Oei, Reed and Schaeffer, Luke and Schulz, Christian and Shallit, Jeffrey},
  title =	{{Decidability for Sturmian Words}},
  booktitle =	{30th EACSL Annual Conference on Computer Science Logic (CSL 2022)},
  pages =	{24:1--24:23},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-218-1},
  ISSN =	{1868-8969},
  year =	{2022},
  volume =	{216},
  editor =	{Manea, Florin and Simpson, Alex},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.CSL.2022.24},
  URN =		{urn:nbn:de:0030-drops-157440},
  doi =		{10.4230/LIPIcs.CSL.2022.24},
  annote =	{Keywords: Decidability, Sturmian words, Ostrowski numeration systems, Automated theorem proving}
}
Document
DOI: 10.4230/LIPIcs.CCC.2018.19
New Hardness Results for the Permanent Using Linear Optics

Authors: Daniel Grier and Luke Schaeffer

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

Abstract
In 2011, Aaronson gave a striking proof, based on quantum linear optics, that the problem of computing the permanent of a matrix is #P-hard. Aaronson's proof led naturally to hardness of approximation results for the permanent, and it was arguably simpler than Valiant's seminal proof of the same fact in 1979. Nevertheless, it did not show #P-hardness of the permanent for any class of matrices which was not previously known. In this paper, we present a collection of new results about matrix permanents that are derived primarily via these linear optical techniques. First, we show that the problem of computing the permanent of a real orthogonal matrix is #P-hard. Much like Aaronson's original proof, this implies that even a multiplicative approximation remains #P-hard to compute. The hardness result even translates to permanents of orthogonal matrices over the finite field F_{p^4} for p != 2, 3. Interestingly, this characterization is tight: in fields of characteristic 2, the permanent coincides with the determinant; in fields of characteristic 3, one can efficiently compute the permanent of an orthogonal matrix by a nontrivial result of Kogan. Finally, we use more elementary arguments to prove #P-hardness for the permanent of a positive semidefinite matrix. This result shows that certain probabilities of boson sampling experiments with thermal states are hard to compute exactly, despite the fact that they can be efficiently sampled by a classical computer.
Cite as

Daniel Grier and Luke Schaeffer. New Hardness Results for the Permanent Using Linear Optics. In 33rd Computational Complexity Conference (CCC 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 102, pp. 19:1-19:29, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{grier_et_al:LIPIcs.CCC.2018.19,
  author =	{Grier, Daniel and Schaeffer, Luke},
  title =	{{New Hardness Results for the Permanent Using Linear Optics}},
  booktitle =	{33rd Computational Complexity Conference (CCC 2018)},
  pages =	{19:1--19:29},
  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.2018.19},
  URN =		{urn:nbn:de:0030-drops-88702},
  doi =		{10.4230/LIPIcs.CCC.2018.19},
  annote =	{Keywords: Permanent, Linear optics, #P-hardness, Orthogonal matrices}
}
Document
DOI: 10.4230/LIPIcs.ITCS.2017.23
The Classification of Reversible Bit Operations

Authors: Scott Aaronson, Daniel Grier, and Luke Schaeffer

Published in: LIPIcs, Volume 67, 8th Innovations in Theoretical Computer Science Conference (ITCS 2017)

Abstract
We present a complete classification of all possible sets of classical reversible gates acting on bits, in terms of which reversible transformations they generate, assuming swaps and ancilla bits are available for free. Our classification can be seen as the reversible-computing analogue of Post's lattice, a central result in mathematical logic from the 1940s. It is a step toward the ambitious goal of classifying all possible quantum gate sets acting on qubits. Our theorem implies a linear-time algorithm (which we have implemented), that takes as input the truth tables of reversible gates G and H, and that decides whether G generates H. Previously, this problem was not even known to be decidable (though with effort, one can derive from abstract considerations an algorithm that takes triply-exponential time). The theorem also implies that any n-bit reversible circuit can be "compressed" to an equivalent circuit, over the same gates, that uses at most 2^{n}poly(n) gates and O(1) ancilla bits; these are the first upper bounds on these quantities known, and are close to optimal. Finally, the theorem implies that every non-degenerate reversible gate can implement either every reversible transformation, or every affine transformation, when restricted to an "encoded subspace." Briefly, the theorem says that every set of reversible gates generates either all reversible transformations on n-bit strings (as the Toffoli gate does); no transformations; all transformations that preserve Hamming weight (as the Fredkin gate does); all transformations that preserve Hamming weight mod k for some k; all affine transformations (as the Controlled-NOT gate does); all affine transformations that preserve Hamming weight mod 2 or mod 4, inner products mod 2, or a combination thereof; or a previous class augmented by a NOT or NOTNOT gate. Prior to this work, it was not even known that every class was finitely generated. Ruling out the possibility of additional classes, not in the list, requires involved arguments about polynomials, lattices, and Diophantine equations.
Cite as

Scott Aaronson, Daniel Grier, and Luke Schaeffer. The Classification of Reversible Bit Operations. In 8th Innovations in Theoretical Computer Science Conference (ITCS 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 67, pp. 23:1-23:34, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)

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@InProceedings{aaronson_et_al:LIPIcs.ITCS.2017.23,
  author =	{Aaronson, Scott and Grier, Daniel and Schaeffer, Luke},
  title =	{{The Classification of Reversible Bit Operations}},
  booktitle =	{8th Innovations in Theoretical Computer Science Conference (ITCS 2017)},
  pages =	{23:1--23:34},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-029-3},
  ISSN =	{1868-8969},
  year =	{2017},
  volume =	{67},
  editor =	{Papadimitriou, Christos H.},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ITCS.2017.23},
  URN =		{urn:nbn:de:0030-drops-81737},
  doi =		{10.4230/LIPIcs.ITCS.2017.23},
  annote =	{Keywords: Reversible computation, Reversible gates, Circuit synthesis, Gate classification, Boolean logic, Post’s lattice}
}
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